Ch7_EigenValuesVector_Ex7.2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0cc2fe52",
   "metadata": {},
   "source": [
    "<b>Question 1</b>\n",
    "\n",
    "Determine the eigenvalues and eigenvectors of $A = \\begin{pmatrix}\n",
    "5&2\\\\\n",
    "-2&1\\\\\n",
    "\\end{pmatrix}$, and plot the eigenspace $E_{\\lambda}$, and write down a set of basis vectors for $E_{\\lambda}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cfd897ee",
   "metadata": {},
   "source": [
    "<b>Solution</b>\n",
    "\n",
    "Solving $det(A - \\lambda I) = 0$ to get eigenvalues\n",
    "\n",
    "$det\\left[\\begin{pmatrix}\n",
    "5&2\\\\\n",
    "-2&1\\\\\n",
    "\\end{pmatrix} - \\lambda\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}\\right] = 0$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "5-\\lambda&2\\\\\n",
    "-2&1-\\lambda\\\\\n",
    "\\end{pmatrix} = 0$<br>\n",
    "\n",
    "$\\Rightarrow (5-\\lambda)(1-\\lambda) - 2(-2) = 0$<br>\n",
    "$\\Rightarrow 5- \\lambda - 5\\lambda  + \\lambda^2 + 4 = 0$<br>\n",
    "$\\Rightarrow \\lambda^2 - 6\\lambda + 9 = 0$<br>\n",
    "$\\Rightarrow (\\lambda - 3)^2 = 0$\n",
    "\n",
    "$\\Rightarrow \\lambda_{1,2} = 3$\n",
    "\n",
    "For eigenvector, solve $(A - \\lambda I)u = 0$\n",
    "\n",
    "$\\left(\\begin{pmatrix}\n",
    "5&2\\\\\n",
    "-2&1\\\\\n",
    "\\end{pmatrix} -3\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}\\right)\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "5-3&2\\\\\n",
    "-2&1-3\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "2&2\\\\\n",
    "-2&-2\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow 2x + 2y = 0$<br>\n",
    "$\\Rightarrow -2x -2y = 0$<br>\n",
    "\n",
    "$y = -x$. Therefore, eigenvector, $u = \\begin{pmatrix}\n",
    "1\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$ and similarly, basis vector for $E_{\\lambda} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3cabb75c",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from scipy.linalg import eig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "b9b5a4b8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.array([[5, 2],[-2, 1]])\n",
    "\n",
    "# Find eigenvalues, eigenvectors, and extracting eigenvectors off it\n",
    "eigenvalues, eigenvectors = eig(A)\n",
    "v1 = eigenvectors[:, 0]\n",
    "v2 = eigenvectors[:, 1]\n",
    "\n",
    "# Create a list of points along the eigenspaces\n",
    "t = np.linspace(-15, 15, 100)\n",
    "\n",
    "# Eigenspace 1: span of the first eigenvector\n",
    "eigenspace1 = np.outer(v1, t)\n",
    "\n",
    "# Eigenspace 2: span of the second eigenvector\n",
    "eigenspace2 = np.outer(v2, t)\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.xlim(-8, 8)\n",
    "plt.ylim(-8, 8)\n",
    "plt.grid()\n",
    "\n",
    "\n",
    "# Plot the eigenspaces\n",
    "plt.plot(eigenspace1[0, :], eigenspace1[1, :], label=\"Eigenspace 1\", color=\"blue\")\n",
    "plt.plot(eigenspace2[0, :], eigenspace2[1, :], label=\"Eigenspace 2\", color=\"green\")\n",
    "\n",
    "plt.axhline(0, color='black', linewidth=1)\n",
    "plt.axvline(0, color='black', linewidth=1)\n",
    "for x_val in range(-8, 9):\n",
    "    plt.plot(x_val, 0, 'k+', markersize=5)\n",
    "    plt.text(x_val, -0.5, f'{x_val}')\n",
    "\n",
    "for y_val in range(-8, 9):\n",
    "    plt.plot(0, y_val, 'k+', markersize=5)\n",
    "    plt.text(-0.5, y_val, f'{y_val}')\n",
    "\n",
    "plt.title('Eigenspaces for Matrix A')\n",
    "plt.legend(loc=\"upper right\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "396b2023",
   "metadata": {},
   "source": [
    "--------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84c86c18",
   "metadata": {},
   "source": [
    "<b>Question 2</b>\n",
    "\n",
    "Determine the eigenvalues and eigenvectors of $A = \\begin{pmatrix}\n",
    "-12&7\\\\\n",
    "-7&2\\\\\n",
    "\\end{pmatrix}$, and plot the eigenspace $E_{\\lambda}$, and write down a set of basis vectors for $E_{\\lambda}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3b807258",
   "metadata": {},
   "source": [
    "<b> Solution</b>\n",
    "\n",
    "Solving $det(A - \\lambda I) = 0$\n",
    "\n",
    "$det\\left(\\begin{pmatrix}\n",
    "-12&7\\\\\n",
    "-7&2\\\\\n",
    "\\end{pmatrix} -\\lambda\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}\\right) =0$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-12-\\lambda&7\\\\\n",
    "-7&2-\\lambda\\\\\n",
    "\\end{pmatrix}= 0$\n",
    "\n",
    "$\\Rightarrow (-12 - \\lambda)(2 - \\lambda) - 7(-7) = 0$<br>\n",
    "$\\Rightarrow -24 + 12\\lambda - 2\\lambda + \\lambda^2 + 49$<br>\n",
    "$\\Rightarrow \\lambda^2 + 10\\lambda + 25 = $<br>\n",
    "$\\Rightarrow (\\lambda + 5)^2 = 0$<br>\n",
    "\n",
    "$\\Rightarrow \\lambda_{1,2} = -5$\n",
    "\n",
    "Now, to get eigenvectors, we need to solve $(A - \\lambda I)u = 0$\n",
    "\n",
    "$\\left[\\begin{pmatrix}\n",
    "-12&7\\\\\n",
    "-7&2\\\\\n",
    "\\end{pmatrix} -(-5)\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}\\right]\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-12+5&7\\\\\n",
    "-7&2+5\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-7&7\\\\\n",
    "-7&7\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -7x + 7y = 0$<br>\n",
    "and,$-7x + 7y = 0$<br>\n",
    "\n",
    "$\\Rightarrow x = y = s$ where $s$ is any real number. The Eigenvector and basis vector for $E_{\\lambda} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "922baf5b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.array([[-12, 7],[-7, 2]])\n",
    "\n",
    "# Find eigenvalues, eigenvectors, and extracting eigenvectors off it\n",
    "eigenvalues, eigenvectors = eig(A)\n",
    "v1 = eigenvectors[:, 0]\n",
    "v2 = eigenvectors[:, 1]\n",
    "\n",
    "# Create a list of points along the eigenspaces\n",
    "t = np.linspace(-15, 15, 100)\n",
    "\n",
    "# Eigenspace 1: span of the first eigenvector\n",
    "eigenspace1 = np.outer(v1, t)\n",
    "\n",
    "# Eigenspace 2: span of the second eigenvector\n",
    "eigenspace2 = np.outer(v2, t)\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.xlim(-8, 8)\n",
    "plt.ylim(-8, 8)\n",
    "plt.grid()\n",
    "\n",
    "\n",
    "# Plot the eigenspaces\n",
    "plt.plot(eigenspace1[0, :], eigenspace1[1, :], label=\"Eigenspace 1\", color=\"blue\")\n",
    "plt.plot(eigenspace2[0, :], eigenspace2[1, :], label=\"Eigenspace 2\", color=\"green\")\n",
    "\n",
    "plt.axhline(0, color='black', linewidth=1)\n",
    "plt.axvline(0, color='black', linewidth=1)\n",
    "for x_val in range(-8, 9):\n",
    "    plt.plot(x_val, 0, 'k+', markersize=5)\n",
    "    plt.text(x_val, -0.5, f'{x_val}')\n",
    "\n",
    "for y_val in range(-8, 9):\n",
    "    plt.plot(0, y_val, 'k+', markersize=5)\n",
    "    plt.text(-0.5, y_val, f'{y_val}')\n",
    "\n",
    "plt.title('Eigenspaces for Matrix A')\n",
    "plt.legend(loc=\"upper right\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39bc3eeb",
   "metadata": {},
   "source": [
    "---------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e5ef4ad7",
   "metadata": {},
   "source": [
    "<b>Question 3</b>\n",
    "\n",
    "Let $A$ be a $2$ by $2$ matrix. Show that the characteristic polynomial $p(\\lambda)$ is given by :\n",
    "\n",
    "$p(\\lambda) = \\lambda^{2} - tr(A)\\lambda + det(A)$\n",
    "\n",
    "where $tr(A)$ is the trace of the matrix $A$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "adec178e",
   "metadata": {},
   "source": [
    "<b> Solution</b> \n",
    "\n",
    "If our $2 \\times 2$ matrix is $A = \\begin{pmatrix}\n",
    "a&b\\\\\n",
    "c&d\\\\\n",
    "\\end{pmatrix}$, then<br>\n",
    "\n",
    "$det(A - \\lambda I)$<br>\n",
    "$\\quad = det\\left(\\begin{pmatrix}\n",
    "a&b\\\\\n",
    "c&d\\\\\n",
    "\\end{pmatrix} - \\lambda \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}\\right)$\n",
    "\n",
    "$\\quad = det\\begin{pmatrix}\n",
    "a-\\lambda&b\\\\\n",
    "c&d-\\lambda\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\quad = (a - \\lambda)(d - \\lambda) - bc$<br>\n",
    "$\\quad = (\\lambda -a)( \\lambda -d) - bc$<br>\n",
    "$\\quad = \\lambda^2 - a\\lambda - d\\lambda + ad - bc$<br>\n",
    "\n",
    "$\\quad = \\lambda^2 - (a+b)\\lambda + (ad - bc)$<br>\n",
    "\n",
    "We know that,<br>\n",
    "tr(A) here would be $(a+b)$ and $det$ would be $ad - bc$\n",
    "\n",
    "Therefore, characteristic polynomila $p(\\lambda)$ or $det(A - \\lambda I)$<br>\n",
    "\n",
    "$\\quad = \\lambda^2 - tr(A)\\lambda + det(A)$<br>\n",
    "\n",
    "Hence, proved!\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b30dc16",
   "metadata": {},
   "source": [
    "------------------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64004a0e",
   "metadata": {},
   "source": [
    "<b>Question 4</b>\n",
    "\n",
    "Let $A$ be a 2 by 2 matrix with $tr(A) = 2a$ and $det(A) = a^2$ where $a$ is a real number. Show that this matrix has an eigenvalue of $a$ with multiplicity of 2."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d0adad0",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "If our $2 \\times 2$ matrix is $A = \\begin{pmatrix}\n",
    "a&b\\\\\n",
    "c&d\\\\\n",
    "\\end{pmatrix}$, then<br>\n",
    "\n",
    "$det(A - \\lambda I)$<br>\n",
    "$\\quad = det\\left(\\begin{pmatrix}\n",
    "a&b\\\\\n",
    "c&d\\\\\n",
    "\\end{pmatrix} - \\lambda \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}\\right)$\n",
    "\n",
    "$\\quad = det\\begin{pmatrix}\n",
    "a-\\lambda&b\\\\\n",
    "c&d-\\lambda\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\quad = (a - \\lambda)(d - \\lambda) - bc$<br>\n",
    "$\\quad = (\\lambda -a)( \\lambda -d) - bc$<br>\n",
    "$\\quad = \\lambda^2 - a\\lambda - d\\lambda + ad - bc$<br>\n",
    "\n",
    "$\\quad = \\lambda^2 - (a+b)\\lambda + (ad - bc)$<br>\n",
    "\n",
    "We know that,<br>\n",
    "tr(A) here would be $(a+b)$ and $det$ would be $ad - bc$\n",
    "\n",
    "Thus, characteristic polynomial $p(\\lambda)$ or $det(A - \\lambda I)$<br>\n",
    "\n",
    "$\\quad = \\lambda^2 - tr(A)\\lambda + det(A)$<br>\n",
    "\n",
    "We have been given $tr(A) = 2a$ and $det(A) = a^2$, therefore, \n",
    "\n",
    "$\\quad \\lambda^2 - tr(A)\\lambda + det(A)$<br>\n",
    "$\\quad = \\lambda^2 - 2a\\lambda + a^2$<br>\n",
    "$\\quad = (\\lambda^ - a)^2$\n",
    "\n",
    "$\\Rightarrow \\lambda_{1,2} = a$\n",
    "\n",
    "Therefore, eigenvalue $a$ with multiplicity of $2$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71e91ab1",
   "metadata": {},
   "source": [
    "-------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d79c588e",
   "metadata": {},
   "source": [
    "<b>Question 5</b>\n",
    "\n",
    "Determine the eigenvalues and eigenvectors of $A = \\begin{pmatrix}\n",
    "1&0&0\\\\\n",
    "-3&1&0\\\\\n",
    "7&9&1\\\\\n",
    "\\end{pmatrix}$. By using appropriate software, plot the eigenspace $E_{\\lambda}$ and write down a set of basis vectors for this eigenspace."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e528d892",
   "metadata": {},
   "source": [
    "<b> Solution</b>\n",
    "\n",
    "We know that, <br>\n",
    "If a $n \\times n$ matrix $A$ is diagonal or triangular matrix then the eigenvalues of $A$ are the entries along the leading diagonal\n",
    "\n",
    "Here, $A = \\begin{pmatrix}\n",
    "1&0&0\\\\\n",
    "-3&1&0\\\\\n",
    "7&9&1\\\\\n",
    "\\end{pmatrix}$ is a lower triangular matrix, therefore, its eigenvalues are all 1. ie., $\\lambda_{1, 2, 3} = 1$\n",
    "\n",
    "Now, to find eigenvector we solve $(A - \\lambda I)u = 0$\n",
    "\n",
    "Here, <br>\n",
    "\n",
    "$(A - \\lambda I)u = 0$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1-1&0&0\\\\\n",
    "-3&1-1&0\\\\\n",
    "7&9&1-1\\\\\n",
    "\\end{pmatrix} \\cdot\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "0&0&0\\\\\n",
    "-3&0&0\\\\\n",
    "7&9&0\\\\\n",
    "\\end{pmatrix} \\cdot\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -3x = 0$<br>\n",
    "and, $7x +9y = 0$\n",
    "\n",
    "$\\Rightarrow x = 0$ and $y = 0$\n",
    "\n",
    "Then, $z$ can be any non-zero value $z = s$, and we can write eigenvector $u$ as $u = s\\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$. Similarly, basis vector for $E_{1} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "c41a4280",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAFUCAYAAADI2uyvAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAAAsTAAALEwEAmpwYAAC6NUlEQVR4nOx9d5hkZZ3ueyp0he7qqq7QVd1dnbsn50A0IKCzXBERBmUVBdSVZVevYa+KgiymBVT0XnWVZQ2ArswAKrKIKKjgigwTmGFgZnpC58qxK+c694+e7+NUdVVXOhVm5rzP4yPT3XXOV+n9ft8vvC/DsiwECBAgQMC5AVGzFyBAgAABAhoHgfQFCBAg4ByCQPoCBAgQcA5BIH0BAgQIOIcgkL4AAQIEnEMQSF+AAAECziFImr2AZSD0kgoQIEBALphaLyBE+gIECBBwDkEgfQECBAg4hyCQvgABAgScQxBIX4AAAQLOIQikL0CAAAHnEATSFyBAgIBzCALpCxAgQMA5BIH0BQgQIOAcgkD6AgQIEHAOQSB9AQIECDiHIJC+AAECBJxDEEhfgAABAs4hCKQvQIAAAecQBNIXIECAgHMIAukLECBAwDkEgfQFCBAg4ByCQPoCBAgQcA5BIH0BAgQIOIcgkL4AAQIEnEMQSF+AAAECziEIpC9AgAAB5xAE0hcgQICAcwgC6QsQIEDAOQSB9AUIECDgHIJA+gIECBBwDkEgfQECBAg4hyCQvgABAgScQxBIX4AAAQLOIQikL0CAAAHnEATSFyBAgIBzCALpCxAgQMA5BIH0BQgQIOAcgkD6AgQIEHAOQSB9AQIECDiHIJC+AAECBJxDEEhfgAABAs4hCKQvQIAAAecQBNIXIECAgHMIAukLECBAwDkEgfQFnNNgWRaZTAYsyzZ7KQIENASSZi9AgIBGgxB9MplEKpUCwzCQSqVgGAbZbBYymQwi0WI8xDBMk1crQAC/ECJ9AecEWJZFOp1GNBpFMBjE1NQUZmZmwDAMJBIJJfc9e/YgmUwiHo8jkUgglUohk8kgm80KpwEBZwWESF/AWYv8iJ6QNoniASCbzcLr9aKjowMKhQIMw0AkEtG/TafT9G/J78j/GIYRTgICzjgwLRy9tOzCBLQuliN6QtCZTAanTp1CIBAAAHR1dSEejyMejyMWi2F8fBxarRYdHR05pE6uxf3OkA1ALBYLm4CARqDmD5hA+gLOeJDUTSqVWpbovV4vXC4XotEoZDIZ2tvbMTo6ColEArFYDJZl8de//hUDAwNYWFhAOByGXC5HV1cXurq6oFarc04JhTYBhmEgFovpZkB+JkAAT6j5wySkdwSckViO6AnZ5hO9TqfD4OAgOjo64HA4aBGXgBD2wMAABgcHAQCxWAx+vx9WqxXHjh2DWCymm0BXV1dOPYCbEpqdnYVWq0VnZyfdBMhJQNgEBDQTAukLOGOQT/THjh3DqlWrChK90+lELBbLIfp8gi8HCoUCCoUCvb29AIBkMgm/3w+v14vJyUmwLAu1Wg2tVouuri7IZDIwDIN4PI5MJgNAqAsIaC0IpC+gpZFP9NlslhJnNBqFWCxGJpOBx+OBy+WiRD80NLSE6PlAW1sbjEYjjEYjgEVCDwQC8Pv9mJ+fRzKZRGdnJ+LxODQaDQAsSQlls1m6IZDfC3UBAY2CQPoCWg5cok8mk2BZlhK9RLL4kSW/f+211xCLxaDX6zE8PIz29vaaSLPSx0okEuh0Ouh0OgCL3UChUAgTExOYm5vD5OQk2tvbaTpIpVLl1BpYlqXPl5wIuCcB7t8KEMAHBNIX0BIgxEe6bgi4pJdOp2mOPhaLIZvN8kL0hdZS7fVEIhHUajVUKhV6enqg0WgQiUTg9/sxOzuLUCiEtrY2aLVaaDQaaDQaGuGTe5PXgvzb4/Ggp6dHqAsI4AUC6QtoGiol+ng8Dp1OR4l+//796OjoqOn+9QbDMOjo6EBHRwf6+/sBAPF4HH6/H06nEydOnADDMPQkoNFo0NbWltN1NDU1BYPBkHNNoS4goFoIpC+goWBZlubnZ2ZmaARbLtHzRW7NJEm5XI6enh709PQAAFKpFBYWFuDz+TA9PY1MJgO1Wk3bRAnJE5BZBKEuIKAaCKQvoO7gEj03ovd4POjt7YVYLEY6nYbH44Hb7c4h+loi+TMFUqkUBoOBRvOZTIYWh61WK4LBIF599VV0dXVBq9Wivb19ySYg1AUElAuB9AXUBcWIPr+I6XQ64fV6kUgkzimiXw5isRharRZarRaDg4PYv38/hoeH4ff7cerUKYTDYSiVSpoS6uzsLFocJuAOjQl1gXMbAukL4A2E6JPJZA7h5KduSHtlOBxGLBbD6Ogo2tvbm7LeeoLPVFRnZyc6OzsxODgIlmXp0JjFYkEwGIREIsmpCxQaGltYWMDc3BzWrFkj1AXOYQikL6AmZLNZWowlRE8IpRDRJxIJ6PV6jI6O4uTJkxgYGEBbW1vD130mEVz+WhmGgVKphFKpRF9fHwAgkUjA7/fD7Xbj5MmTAACNRkM3AplMBmBxAyCCckJd4NyEQPoCKgYhep/PB7lcDqB8om9GRF8JWo3oyj2NyGQymEwmmEwmAIuv/8LCAvx+P+bm5pBOpyGXy5FKpRCNRqFQKIS6wDkKgfQFlIVsNkvz8+l0GizL4siRI9i+fXvVRM8wTEtq1POxpmY/L4lEAr1eD71eD2Dx/bPZbJifn8fExASi0Sg6Ojpyhsa40X2puoAgJnfmQiB9AUVRiOjzc8CZTAZutxtut7viiF4gjNKoZVCMC5FIhI6ODnR2dmLt2rVgWRbhcBh+vx/T09MIh8OQyWS0Q4gIxRUSkyP/drlctPtKKA6fORBIX0AOCNEnk0nqHZt/vE+lUvB4PIhGozh48GBNqZtmRsREBycQCFATlbMZ3A2EYRioVCqoVCoMDAwAeENR1GazLVEU1Wg01FISWPyczMzMwGg00roAIX2hLtDaEEhfQEVE73K5kEwmYTAYIJfLsX379qrv26z0DsuyiEQidC5ArVYjkUggkUggHo/DarVCr9dTJ61mg881LHetQoqiZGiMqyhK2kQLDY0JdYHWh0D65ygqJfpUKgW9Xo/x8XEolUoAgNvtrnkdjSJ9lmURCATgcrng9/upicrq1ashFoshkUiQzWbxt7/9Del0mua9VSpVzlBUJaTFB8Hx+fpUeq22tjZ0d3eju7sbwOLQGCkOz8/PIxQK4fDhw/Q0QF4foS7Q2hBI/xwCl+idTicAwGAwVET0fKLeX3aWZREKheByueDz+aBSqdDd3Y2xsTF4vV6Ew+GcNYhEIkilUgwODmJkZITmvX0+H06ePIlIJIL29nY6OFUP6eZC4OsetdYHxGIxVRRNpVJ45ZVXMDg4CL/fn/P6LKcoCiz1FxCGxhoLgfTPcuRH9MDiF41bmG0k0XNRr/ROOBymk77t7e0wGo0YGRnJSUWUuz6S9yZDUZFIBD6fD1NTUwiFQnQylhQ/+SYsvl8fPjcQoiiqVqsxNDQElmURjUZpm2gwGERbW1tOXaBYcXh2dhadnZ3QarUFNwIB/EEg/bMQxYg+P+paWFiA1+ttKNHXC9FoFE6nEx6PB3K5HEajEUNDQxCLxUUfU4xQi/2cq5g5MDBAJ2N9Ph+VTSaeuvF4HNlslpfnxhf4ThUVGhprb29He3s7zGYzgMKKotyhMaIomkwmc66dL90hFIf5g0D6ZwnKIfpUKkXbK6PRKJRKZVOJvtZIP5vNYnZ2Fm63G1KpFEajEZs3b6ZGK6XuXSu4k7GE5MgmEI1G8frrr0Mul1MrRbVavewmVM+1Avy1f1ZyrWKKon6/HzMzM1RRNBqN5pyUljOZEeoCtUEg/TMYmUyGSiCUInqXy4V0Og2DwYDx8XGEQiHE4/GaCb9WIqmU9BOJBFwuF5Vdlkgk2LhxI6RSadVr4BMKhQJ9fX3w+/3o7++HTCaD3++H3W7HxMQE1cghJiqlNoFWTu9Uc618RVHSMjsxMYHZ2VlMTk7S4nlXVxetmwh1Af4gkP4Zhkwmg1gshlgsRomuHKJfuXJlTh96OBxu+tRouV/MVCpFiZ5lWXR3d2P9+vU4fPgw1eNvVeRHuslkEj6fDy6XCydOnIBIJIJGo6GngUKnFD6Jmi/wOTRGWkD7+vqgVqtp8XxychLhcBgKhYJuAmq1uqzicDKZhFwup8JzwibwBgTSb3GQASLuZGwwGITL5cLKlSsrInou+CiikmtU+4Vabg3pdJo+n1Qqhe7ubqxevZpq/ZypaGtry9HISaVS8Pv98Hq9OHXqFADQk0BXV9dZF+kvdz1C5vnFczI0ZrVacezYsWUVRcm1Xn/9daxZs4YKzQl1gTcgkH4Lgkv0JHVDPqikpxzIJcZyiJ4LPkmfL2QyGdpFFI/HaSqqWAqq1b+45axPKpXm9MKn02n4/X4qj0By2Q6HA1qttiZFUr5z+nwim80WXFshRdFkMllSUTSbzdKUTzl1gVb/LPEJgfRbBKWInnwok8kkPB4PPB4PIpEI9Hp92UTPRSuQPle7x+Vy0edTiZFKLfdvdnqrECQSSU7OOxgM4tixYwiFQlQtU61W05NAJScfvp8vn2m1SjaktrY2GI1GGI1GALmKovPz80gmk0gkEnA4HNDpdFAqlUJdgAOB9JuIfKLPZrP0mJtP9IQYs9ksNdTYsGFD1fduJulns1kazfp8PhgMBgwMDDRs2AkoHoW32hddLBZDJpNhfHwcwOJrRwjOYrEgmUxSaQStVrvs5t/KkX4tayukKPq3v/0NqVQKx48fRzQapUNjWq22oKIosHQTOFtNZgTSbzCWI3ruMTOf6A0GA1atWgWFQoFoNIqpqama1sFXaqbca5C5AJfLhYWFBTq2bzab0dXVVdW96zXc1crSyiKRiE4Ej46OIpvNIhgMwufz4ejRo4jH41CpVPQkQKJcAr6Hs/gCnxsSCZpGRkboZyQSidA20VAoRBVFua20+XWBbDZLu+ICgQAUCgWUSuUZXxcQSL8B4BK9x+NBNpulI+rlEj0XrZCaIddYDizLIhgMwul0wu/3Q61Ww2g0YsWKFWAYhuZjBRRHKTIk3T8ajYb+fTAYhN/vp1EuaYHkRrJ8rItP1HMT4Q7V9ff3A3hDUZS00orF4py6ALczDgAcDgcMBgPa2trOeDE5gfTrhGIRfTgcRjabpdFtIaIv1aXSSqSffw3yHJ1OJ3w+Hzo6OmA0GjE2NlbwS92KefUzGQzDLJFGIC2QDocDiUQCgUCgZv0gvrt3ihVy64V8RVHSRUUK6JlMJmcTYFmW5vyBM9t8XiB9HlFO6kYsFiOZTMJqtVZE9FyIRKKaR/z5Jv1IJEL1bhQKBYxGI4aHh5cdPqr1C1HrczgTNpxayZXbAimRSKjkRjH9IHICrfe66n29SpHfRZXJZBAIBGiraCAQQDQahdFoLKooCpwZdQGB9GsEy7J0MjaRSNA3n+QVCZLJJFwuF6xWKzKZDMxmc9V9560S6WezWVgsFipVbDQaMTAwUJYMAl9rqBbky0e8ftvb26FWq+tyj1YBIdZS+kEymSxHOqLYCa0effqtArFYTE9DAPDqq6+iu7sbiUSCKoqSzZIMl+WneAg3tJr5vED6VSCf6MnRlKsFArxB9G63G9lsFt3d3TCbzUgmk9StqBo0k/Tj8Th9TrFYDF1dXdi0aVNTZBBq6R5aWFiA2+2G3++HXq/HwsICLYTOzMzAYDDQL/LZgmIiaYX0g7jDUFKplJ4ESNHzbIv0S4FlWahUKvT09CxRFJ2fn0cwGFzyOhUaGiMpoVQqhbm5OQwPDze8LiCQfpkgRM9N3RCi50a2iUSCipqxLLskdePxeJBIJGpaC8MwDU3vkM3L5XIBAIxGI9avX48TJ05Ar9dXTfiNjPRJ95DD4aApDbVajTVr1uS8h3v27IFMJqNfZCKYVi/p5HLWzec9y7lWfr6bKGU6HA5a9JTL5ZTAyj3ZLYdG5/QrRf5JhGGWKoomEgn4/X4qsQGAngSIoih5bDqdhtPpxNDQEE0JyWQygfSbjXyiP3HiBMbHx4sSPSHF5XL0fOTjyZRhLShFuFxZh0wmg+7ubqxdu5aOtZNrNBPldA+Fw2E4HA74/X50dnbCZDJh1apVdEYgH2KxGCaTKcc3Nl86uZxNoBXrBdVuIIX0g2ZmZuDxeLBv376y9IPqtbZGIZvNljz1yWSyHIkN7nQ1URTt7OykQnLcPH8jZbgF0s9DPtFzc/ShUIjm6QsRfT4pFkIjWiXLvUb+OtLpNDweD5xOJ1KpVN31e/h4LQo9nqutr1AoYDKZMDo6uuRLW869iWomkQCIRqM5m4BCoaCbABn64RN8D1Txka5qa2uDRqMBwzAYHx/P0Q8iXrpcJdFypSPOdNLPR/50NVEUJebzgUAAhw4dypGPaAQE0scbRJ9MJpFKpXKInvtGk8JlpUTPBR+RPh8gp4VMJgOv1wun04l4PA69Xo+xsTG0t7eXvEYrkD4BSUE5nU6IxeKS2vrVEgw3/02KoF6vF9PT0wiFQtROkfs5ahXwvYGQa5XSD8pmsznSEY0iNz5RDenngyiKdnV1wWQyYWJiAqOjo1RHSKfT8bTa5XHOkn65RM+N6OPxOFiWrZjouWhmxwoBkUEIBoM4cOAAdDodhoeHqzL+buZzYVmWzgOQFNT69etrEiWrBNwiaH9/Py3uEf/dQ4cO0elYnU5X8etbrzXzgeU2kPwIlxiq+3y+HP0gchrgWzm1Hp9JPkg//3pisZi20wrpnTqhEqInHSoAaD778OHDdKKvWjQr0icdK06nE6FQCGq1GjKZDNu2bauaCJoR6WezWTpoFAgE0N7ejhUrVrSEzSO3uLewsIChoSGIRCL4fD6cOnUK4XAYHR0dNB1UzibQqno5layLa6gOvJHm8Pl8eO2115BMJhGPx2G1WkvqB/G9tmZdk+9NpBKc9aRfK9HnR/R8DMs0smMlEAjA6XRiYWEBGo0GPT09WLVqFTKZDA4fPtzUyLPc14I8D0L0Wq0Wg4ODyGazMJvNNZNEvSASiZb0xJPpWNLrTU4CWq12iU5OPdCISL8UuGkOoh/017/+FYlEoiz9oFKoF6HyTfrVWGfygbOW9F977TWMjo4ilUrh6NGjWL16ddVET0Ci9FrerHpH+izLIhQKUb0blUoFo9FIu44I+Coo1/O5kM4bn89HnwfXOKbWL2Gh51/ymh4P2oaGAI0GSYulovtxp2OJQUg4HIbX683RydHpdDTirVcevpWuRbrhRkZG6LXz9YMqOSG12qBXIRDpdC4aFYCdtaT/wQ9+EM8//zxEIhFisRhtj+ISPcMwFRVj+SBKPtot80FUBIkMQnt7O4xGY8GOFYJW6CIqtIZYLAan0wm32w25XA6TyVRSzqHae1cD0d69YNJpsMEgL2sgmwAZ+AmFQvD5fJiYmEAsFqOGIMTIvhY0K71TKZbTDyJpskJSyQSt3vMPCJF+3cAdc56bm4PH44FIJILBYMC6desqLvjxpXnDV3TMbU2Uy+UwGo0YGhoq68PUKlIOLMtS0Tmn0wmGYUp23vB5/2JrKnrPl18GC4BJJoHZWWBwsOJrFL02w1CvBEJ2VqsVFosFx44dQzweR2dnJ414q0lrtWKkXwqFTkhEKpl0TXFbZ6VSactH+kJOvw5QKBT42te+hk2bNlHZ2Vo7O1pB6CwWi1GXqZMnT5ZNkIXWUStqeS6ZTAahUAihUAg2m61kaq1VIHr5ZUChAGIxiP78Z2Rvuqlu9yLdQWq1GqtXr0Y2m6UngSNHjiCRSNBWyHK6YFo1vVMpuPpBpGuKDNHNzc1hYWEBqVQKU1NTy+oHNRMC6fOEQCCAn/zkJ/jVr36FqakpXHrppbjgggtgt9thMplqbuXja5q20muQlJTL5YJYLEZ3dzeUSiU2btxY01pqRaWkT1pFHQ4HIpEIZDIZuru7MTw83JD714xMBsz+/YBGAzadrjvp50MkEtG0x/DwcI6BCumCWW4T4Du90ypEmq8fFA6HceLECcjl8mX1g5oJLuk3uu257qTPMMxPAFwJwMWy7LoCv2cA/D8A/wtAFMBNLMu+AgAPPfQQvva1rwEA7rjjDtx4443L3iudTkOpVOKXv/wl3v/+9+Pmm29Gd3c3nE4nLymVRqZEuCkPAEt60K1Wa0uMrpd6LqQo53A4qGNWf38/VCoVbDZbg1a5FNW8bszRo2DCYWT7+sCcJn2wLFDH92C595hroDIyMrKkFTKVSuVsAsCZmd6pFCzLQiKRoLe3l+oHJRIJ2upL9INId5BGo+FFP6gSFIr0z6ZC7oMAvg/g4SK/vwLA+On/nQ/ghwDO9/l8+PKXv4z9+/eDYRhs3boVV1111bLWejqdDrfccguARb2QeDwOgL+OmXqnd9LpNCX6TCYDg8GANWvWFDy2k+s0u+WyGCKRCBwOB7xeLzVSKdRB1Mzp5Eo3cGbPnsX/UKnAZrMQnTwJ5sgRsOuWxDJNQaFWyEAgAK/Xi/n5eYTDYQSDQWQyGWi12ppOvq06PwAUJlSZTLZEP4hMwp48eRIMw9CTQL5+UL2GvbinjUZ+j+tO+izL/oVhmKFl/uTdAB5mF1/ZPQzDaBiG6fnFL36Bt7/97TRCefvb345nnnkGf//3f1/WfeVyOZLJJIDWI30uMpkM1btJJBIl9W6412n2ZG/+GojsssvlQltbW0WFZT7uz9c1i0H08stgDQZALqfRvejPf0amzqTPRz88ABw7dgxyuRyhUIhOxhKhtGo2gVY9NZRzPfL5NBqNAN5wzvL5fFQ/iLw29agJZLPZpsiRA62R0+8DMM/5twVAn9VqzZl+NZvNsFqtZV9UJpO1ZKQPLH4oSUQfjUah1+sxOjpalt4NAYmSm5mbJBKxNpsNTqcTLMvCaDRi48aNZX2gm71xkaEvt9sNhUIBnU5HNc8LgXn5ZWTPOw9MMAjI5ciOjYH505+AT3yi4LX5IDI+Xx/SHUQmY7nyCLOzs8hmszmbwHLvIZ9tkXzXB6q5XiH9IPLakA6ho0eP8qYfJLRs1gFyuZzq1vNF1rWQFCliEqIPBAIYHBys2qO0Hv3+5YKItDkcDmQyGfT19VXlAtYswbZYLAa73Q63241MJgOTyYRMJoPp6Wn4/X4cOXIEBoMBOp3ujWlQrxeikyeR/uAHwTz7LACAfdvbIHrkESCVApoUtdWCfHmETCZDo92ZmRnq5UyIrl6RaT0kDmq9nkQigV6vh16vRzKZxKFDh2A0GuH3+3nRD8pkMud0944VAFfQxgzA2tfXh+eff57+0GKx4JJLLin7onK5HLFYDEDzIn2WZaneTSAQQFdXF/r6+hAOhzE2NlbTWhodJbMsSztvwuEwJYv29naaJ21lpFIpqsJJhn/0ej2d1JZIJDCbzdi/fz+GhoYQDodx/PhxxGIxqFQqmA8fRg+A7PnnQ3Sa9LOXXgrxf/4nmH37wF50UV3W3cg2S7FYTIkOeCPa9Xq9mJqaopLJOp2OV9Lie5iqXuJopfSDyAwFkY5o5BorQSuQ/pMAPs4wzC4sFnIDLMvafT4fvvjFL8Lv9wMA/vCHP+Duu+8u+6L1iPTLuQ7pViEyCBqNhnf5AHKNehdByYQoMSHRaDTo6+uj5iHz8/MtI61cCFxxtlgshu7ubloYDwaDBbuHiGiaTqejg0DBYBCin/wErEiEl9JpbIlGIZVIkLrwQkgYZjGvXyfS5xOVbiDcaBd4QzLZ5/PRlJ7f76/JPKWadTX6eoUIulDRnEhHkEE6lUpFTwL5+kH51zyrCrkMwzwC4BIAeoZhLAD+FYAUAFiWvR/A01hs1zyFxZbNmwFAq9XiS1/6ErZv3w4AuPPOO2lRtxw0snuHjIkTmV/SrTI2Nla33bye6Z1oNAqHwwGPx0MlHQo9l3rIMNT6+PyNioizVZtGI6cC6YkTYNevx3lvextEbW1Ip1I4ODuLzePjYH77Wyx87GM1EV+pNbTCtbiSyaQ+IBKJ4PV6cerUKTAMQ+sBXV1dZees+c7p1yPSL3U9bvvs8PBwjqTGiRMnEIlEqH5QV1fX2Z3eYVl22Xab0107/1zodx/+8Ifx4Q9/uKr7NiKnT/RuCDl2d3djZGSkIW8m31EyGQBzOp2QSqUwGo3YunXrsl/cZhdiueBq9iiVSphMJv423UwGzL59yP7930MkEkEqkUAqkeD8888H8853Qvr972N6fh6nTp2CSCSCVqulPsq1gu+BKj6vRU4CpPhJOmBIGyR5LYiDVrHPEt/pnVbYRApJaoTDYfj9fkxOTsLj8SAej8NgMKCrqwudnZ28rbcUWiG9UxfUq3snHo9jZmaGCoJ1d3djcHCw4ZV4vnRv7HY7HA4HWJZFd3c3NmzYUJG9XS2va63PgWVZeDweGmWaTKaqJClKgTl2DEwohOz55y/95WWXgfnOd7DK7UZ2xw4kk0n4fD7Y7Xa8+uqrkMlkNBfcDGN1LupdH8jvgCG98MQovNgm0IqF3Pzr1bqJcPWDBgYGcPDgQZjNZsono6OjvJvJFMNZS/r5ffrEcb4akP5zq9UKhmEwODhYE7nwMVhVLeFms1naeRONRpFIJLBq1aqqxbsaHelz8/R+vz8nT18pypVWJkNZbAHSZy+6CGxb22Lr5o4daGtrg8lkgsPhwIoVK2j6g+upyy2Cl/MZaJX0DhfV9MKTDdHpdOL48eOQSCS8mKYUWluzW0DLuaZKpYLBYEBfX19Dg8azlvQVCgUtAlcT6RPPVZfLBYZh0N3djYGBAaTT6Zq7Vfjosa8kp0+6iBwOB0KhELRaLYaHhxGPxzEwMFDTB7oRhVxyNLbb7fD7/ejq6sLAwAAkEglMJlNVhL8cYeWvSbR3L1i9Huzo6NI/VirBXnjhYjG3wLXkcjk1VmfZN+wUiUQwVz+/UMdHK6d3Kt1AyIZoMpkALKYUSUeY1+vFvn376EmgloGoekT6fJ/Q8nP6Z1Uht1moJqefSqXo0FQ2m12i/OhyuZBKpWpeWyM0fAhREhMStVoNk8mEVatW0Q9YrcXgehdy4/E4HA4HHZ7Kz9OTtFS9QYayimnsZN/2NkjuugtwuYDTqY1ipwhip0ictEKhELxeL+34UKvVdBMgn7tWnHzl41oymQwmkwlKpRISiQTj4+Pw+Xw5ImnEX5gUjcsB39Ou9WivPNdbNuuCckk/nU5TGYRUKgWDwVA03dFKk73F0juxWIx23igUimXNVPjonuG7bZToDxFCr1eevmz4fBAdP470+99f9E+yl14K3HUXRC+8gOx115V9aW6xj6hmkt7v+fl5pNNptLW1QSqVIpVK8UJkrUT6BCSSztfHicfj8Pl8sFgsCAaDaGtroyeB5TaBRrRstuI1y8VZTfrFCrlkotTpdCIej0Ov12N8fLzkQAWfpF9rhMq9BklFOZ1OiMXihpmQ8BXp50sud3d3l1VnaERNgdm/HwAWI/0iYLdsQfpznwO7Zk1N98rv/c5kMpiamoLP58OBAwcAoKqWSLrOJqd3Kr2WXC7PUcokmvnz8/MIBoOQy+U5mwC5Rit075RCM8USz2rS50b6XGGzSCQCnU6H4eFhdHR0lH1NPls/a70Oy7I0Ckqn00ukl8tdR7OGq0h+2+/3Y9++fTmSy436MpSzdtGePWBFIrCn50UKQiJB5itf4XFlixCLxVCpVBCLxRgZGclpiTxx4gQkEglNBZWTA2+19E6l11IoFLQ+AryxCZAiOdkEYrFYzdaSXJwJRuuV4KwlfZlMhlgshr1790Iul2NhYQESiaQmYuErsqw20ud2riwsLECj0ZR1QimGZpB+IpGAw+GghjAymQzr1q1r+FG32Puf/3PRyy8vSidXEBzwCe7rm98SSTTirVYrjh49SklPp9MV/Yy3KulX8/5zNwGWfcM9y+/3w+PxwGaz0ZNALcFEPbp38iEUcmtANpvFX//6V3z3u9/FSy+9BAD4/Oc/D5FIVLPeTTMifaIE6XQ6sbCwQCdM29vbqVtQLetoBOnn5+mNRiM2bdqESCQCl8tV9Req7umdbHZxKOu9763fPcpAMULIz4FHo9EcVUgyAUqE41o1vcNHdwzDvOGeFYlEaDcU9/Vob2+nm0AlE9pngtF6JTjrSP/f/u3fMDs7ix07doBhGHznO99BLBaD2+2u+dqNVOsksg5erxcqlQpGoxErVqygHz6/31/3DqByUOzxJP1E8vTLFchbDfQ5sSxSv/41cNpjuVLwQRSVvD9cy0CWXTQP93q9VDiOiM5V2+aav65WPDUAbwikkU4p4qNL2mUnJycRDoepxpJWq112ZqIR2vdnXaTPMMzfYdESUQzgRyzL3pP3++8AeNvpfyoBdJMPu1gsxvr16wEAAwMDePLJJ5e91x133AEAOHnyJH77298CaK2um+Wuw5USkMvly5qQNKLts5zH54O0iXq9Xmg0GpjN5qKTqM2sKZQFsRjsxRfX7/p1BMO8YR5OhOP27t2LZDKZY6VISK9SA5VWSO8sd738tRVql41EIvD5fHRmgpyM8jeBZnba1AONEFwTA/h3AG/HokHKPoZhnmRZ9ij5G5ZlP835+08A2IzTwmsKhQKHDh2q+L71ElzjK6dP1pMv+UtSH6UiCz6KwXyRbiKRgNPphMvlor3X5WgQNfvInEqlMDk5Ca/XS4uiraaZwxe5MgwDiUSCgYEByOVyZLNZKptMDFSIbHI5wnGtHumX89kjmyLZBMLhMHw+H06ePIlIJAKVSgWtVot4PM5rYbjZelWNiPTPA3CKZdkpAGAWJZTfDeBokb//eywqcd5cy03rJbjGFyH4fD7Mzc0hlUotGQIrB/X26y0FYrjhcrkQDAZhMpnKdsziotFfgHQ6TSU1EolEzgbl9Xphs9lw8OBBKJVKKpdQrUJnq4FLrlwdHOAN2WQyLUx+r9PpoNFolpBoq+X0uajm5MDVxiEno3A4DK/XC4/HA7fbDZfLRV8zhUJRU2G4mZ+nRpB+ITvEAspVAMMwgwCGAfyJ/Cwej2Pbtm2QSCS47bbbcPXVV5d1U4VC0VJ2idxedCIlMDY2VpFFIhfNSO+wbK6Rikqlgkajwdq1axty/2ofT9Ztt9sRjUbR3d2NsbEx2Gw2GI1GaqJiMplgt9uxcuVKAMjJ/xKbQZ1OhzPPI+sNFCMbrmwy8IZOjsPhwMTEBKRSaY5wXKtH+nwUhskmkEgkoNPp0NbWBp/Ph4mJCWquQzbGSmpVhU4iZ11OvwJcD+BxlmWpjMns7Cz6+vowNTWFSy+9FOvXr8doIQ2UPMhkMhrp8/WCVuOcxTVU6erqgtlshkKhgEqlqprwgcaSfn6enhiphEKhgkYkrYJoNAq73Q6v1wu1Wp3TrhuJRJZ9LCmKkiIgkUt47bXXsDYYhFQigf905FfPaWG+ybVc5OvkxOPxHOG4eDwOi8WC7u7usoXjlltXsyP95UBIOl8qORgMwufz4ejRo4jH49Q5q5SIXLNrBI0g/YJ2iEX+9nrkaeuTQYyRkRFccsklOHjwYFmkL5FIkMkUksCqHuWSJFdnv5ChysLCAi8TufVM73Dz9IQACuXpm1mILfR4kr5xOBwQi8UwmUxFi+GV3IcrlyDp7KTpkMnJSZoK0ev1OUNSrZYSqoVc84XjXnrpJYhEorKF40qti2+SrneNgGEWzXXUajWV0CCBwZEjR5BIJKBWq+kmwO2WOhdIfx+AcYZhhrFI9tcDWCJkwjDMKgBdAF4iP/P7/VAqlZDJZPB4PHjxxRfxuc99rgFLLozlPkj5JGk0Govq7LdCEbbQNTKZDBWcS6fTMBqNy+bp+ZJhqBWF0jeV1ki4ayr5NwCkEglNA5FUCHdIKhaLIRKJQCaTtVQUzFdRWCwWo7+/n0a9xCVqOeG4YqhVcTYfzZBhEIlEdBMYGRmh9onEQzeVStEUoUKhOLtJn2XZNMMwHwfweyy2bP6EZdkjDMN8BcB+lmVJD+b1AHaxHBY4duwYbrnlFhrV3nbbbVhTo74Jn+CqcpKho3KKmc0uwnKvwa01hEKhsnWI+FpDLUilUnA4HJienl6SvikHZO35z6HS55SfColGozh48CBmZmYwMTEBlUoFvV4PnU5X1UbEF/h8r7jRNPckNDQ0VFA4jtsZlP/9aMWcfv71KiVprn0i2QTIazIzM4NoNIojR45QQ5lGfi4aktNnWfZpLHrhcn92Z96/78p/3EUXXYTXXnutqnsyDENJie98IUkfJJNJGAwGrF69uqJhl1bosY9EIggEAnC73dBqtejp6cmRXW7EGqp5PDd9k0gkoNfrsWbNmoojxXqmXpRKJeRyOT1tBINBeL1eHD58GKlUCl1dXdDr9WW3RvKFRp0aCgnHkfbQqakpALnCcWdKTr8WcF8TvV6Pubk59PT0wOfzIRqNCnaJrQZu10o0GkUoFMLIyEhFYm1cEAG4WlBNiiiZTMLpdMLpdKKtrQ1tbW0YHBykHRvVrKERkX6x9I3dbodSqWy4VWUl4OZ+R0ZGaKsrtzVSp9NBr9cXlQtuxS6ZSt53sVhMO38ALBGOI4NiCoWiJvMUglZoAV0OJJ1FNr5GyzycU6RfyYee5CmdTid8Ph80Gg16e3sRDofLKiQvBz5y+uUOihF1UYfDQfP0xAd3amqq5g9bPSP95bpvynl8K0IsFkOv10Ov1wNY3Ii9Xi8sFgsCgQC1U9Tr9bwOBNUD1X528oXjjh8/DpZlKxKOWw5nYmG4kTirSZ9E1BKJhObRS0WF0WiUmpC0t7cvMSHh482vd06fZXPtEXU6XcGZgHrIMFT6+EZ037Qy2traqGgaVx/mxIkTiEajEIlEUKlU6O7urjnv2+yhoGIgRVCyCZQSjiv1HJox4dvM61WKs5r0yVRuKdJPJBJUCkEqlcJoNGLr1q11Ixo+i7BcRCIR2k+vUqmW2CPyvY5md9/Uev9WOyXk68Nks1lK/q+++ioymQwtiGq12qo+n61I+vkkXUo4jrSH6nS6moXjyoFA+mcQ2traEI/H0d7eviS6JnK/XD9ckvYohVojCT4ifZLe4bpmkanSciPjWtdRK+nG43FEIhHs27evqu6bWrDcPVplMxCJRJDL5ejo6IDZbF4ilUBy5aQe0IqEXg6WS8cUEo4jhXE+hOPKQT1In3w/m/FZO6tJP190LZ1OIxgMwuFwIB6Pw2AwYOXKlRWNUJebJloOtZJlJpOBz+eDz+fD4cOHYTQaK3bN4mMdtXbfkH7vep6qKkWr9cRzr5MvlZBIJOD1ejE3N4dgMEj1gvR6fU3aMI1GJTnz/MJ4IeG4eDwOt9tdVndUueDztcxkMkuuJxRyeYJcLqfFwHA4jMOHD8NgMGBoaKhqES0+SL+aCJtlF81UHA4HgsEgOjs70dHRgU2bNlW9jmZq36xduxYMw+Do0aNVv5ZnYiGXT8hkMuohy02DEG0YEgET3ZhWRS0n50LCcX/729/KFo5rBvL5o9Gb81lJ+izL4tChQ5iensa1116Lf/mXf8HGjRsxMjJScz8sH/LKlZAVt7BM8vQrV65EPB7HyZMnG7aOah5fqvsmlUqdtaTdaGnl/DQIGQYiJwESAXs8nqpM1esJPguvEokEEokEq1atAlBaOK4Zp6FGmLIsh7OO9GOxGN761rdieHgYer0e3/rWt3DBBRfQtrBawVe75XLXSKVStJ9eIpEULCzztfnwYdDORaO7b2ot5JKJZLlcDrVazePKmgvuMNDY2BjS6TRefPFF2hvfCuRHwHeLJRfFhONISoy0yOp0upqF48rFOVPIZUq7Z90E4Js4Lcb2n//5n/joRz8KAHjooYfwta99DcCiM9aNN95Y9D4KhQIvvPACFAoFbrnlFvrzVpBXJigUIWezWdpPn0wm0d3djXXr1hXtYGmFqV7yBWlW900tX1CijbNv3z5otVqwLIujR48imUxCqVSir6+vZfrk+SAiiUQCqVSK1atXA8hVzQwGg+jo6KDk1+jn3cjhpHzhONIiy4dwXLk4J0ifKcM96zR2syz78dP/zQKAz+fDl7/8Zezfvx8Mw2Dr1q246qqr0NXVVfR+pDBbL/csviJ90ongcDiwsLAAnU5X9qRvK4i2xWIxxGKxqrtvGp2TJ6cQu91ONfTJCUoqlYJlWbzyyitgWZYKh5Gx+XpLKBdDvV6ffPIjhiHkeWs0mjf8A+qcimjW/EB+i2wx4bhUKoVEIsGbPs45Qfqo3D2L4ve//z3e/va300LN29/+djzzzDP4+7//+5I3rQfp80FUiUQC4XAY+/btQ0dHB0wmU47peTlodG2BgJu+EYlEEIlETeu+KWf9pABus9kQDofpCYplWZw4cSLny8cwDKRSKXp7e9HR0UFTPx6Ph0oo6/V6jGUykLRQTrxWMMwbhiFEMI3bEcOyLC2GdnV18U5YzZKHyEcx4Ti3241XX321pHBcucgn/bO1kFuue9a1DMO8BcCJubk59Pf3w2q1or//DTl+s9kMq7WYHH8u6mGZWO11iBcuaVWslSwbmd5ZLn2zb9++luy+SSQScDgccDqd6OjoQG9vL9RqNf2CkWBgORBdHKIZQ1ok4/E4Muk0Thw6RCUVCg0J8fFlbkYUnN8Rk0ql4PP54HQ6cfz4cbS1tSGRSCAYDPIyV8E36fN1LaKUKZPJcN555xUUjuNuAuV+D86VSL8c/DeAR1iWTTAMc8uNN9547Z/+9KeSD1oOMpmsqemdbDYLr9dLFSFJlFlrqyLQGNIv1X3TCuCun7zedrsdqVQKJpMJW7ZsKZiWKSVjUQikRVJ6Ws5idHQUHo+HDglxU0FnU1cSmVI3Go0AFtN6L7/8MpVJ4E7IVjLzQsBnIZdvQuVuIqWE4yQSCa0HLCccx7d/QKVoFOmXdM9iWdbL+eePDhw4cD+w6Jz1/PPP019YLBZccsklZd20GZF+oTz98PBwTp4+nU7zko+vFc3WvuGrZTQSidDNSavV1qSAWglISmR4eJgOzHm9Xpw8eRKxWAyzs7MwGo01b5SttMkCizWztrY2bNy4kdYDPB4Pjhw5gmQySesBWq22rBQIn4XcRsoq5wvHJRKJJWY6hYTjMpnMOZHeKemexTBMD8uy9tP/vIp0GezYsQNf/OIX4ff7AQB/+MMfcPfdd5d100ITubWiWAE1FovB4XDA7Xajvb0dPT09RfP0rTJURNZRbfcNH/evFul0GgsLC7QXvaenp6CdY6MgFotzpmX37NlDlUyJsXo1Riqt8DnJB3dN3HoAsQ0kUhHT09MAQKPjYsNRfKZkmqmIKZPJqHgeUFw4LplM5nS+NRqNMlEpxz3rfzMMcxWANADfgw8+CGDRbOFLX/oStm/fDgC48847aa6xFBQKBRYWFgDwG+mTN4rk6Z1OJxiGWTadkH8NPtZSK5LJJPx+f1O0b6oBOUXZbDaEQiEoFAoYjUaMjIw0e2lLIBaL0dPTk6MX4/F4qHAa8dQtpzDaau/HciSdXwdJpVI05Xbs2DHI5XL6ezIVz2d03shIvxSKCceFQiG88sor1EO3t7eXt/WWg4bl9NkS7lksy34BwBe4vyb/8eEPfxgf/vCHK75nPdI7DMMgEAjA5XIhFotVFRU380ucTqfhdDrhcDjAsiykUinWrVvXUhOa+Ugmk7Db7XC5XFAqlejt7cWqVauoR0Crg6sXMzo6inQ6TQujExMTlAj1ev2SAaFWjPSB8j/DUql0iZWk1+vF5OQkPQHF43GqhlsrWlX7njsx7XK5sGHDBjoh3ej3uJUKubxDJpPxQvqkf9fhcMDlcqG9vR1jY2NV6/c0GoXSN+vWrUM8HofT6WxJws9ms/D5fLDZbEgmk+jp6cHmzZub0ivPNyQSSU4uOBqNwuPx4OTJk4hEItBoNLQg3IqohVhJ9Nvf30+/V6+88gqOHj1Ki+GkHlDNe13PQi5fIIVctVoNlUrV8M/0mf8NWgYKhaKm7p1YLAan0wm32w2lUgmTyUSn9FQqFe/r5Rulum8SiUTLRZJkzUQjZrmibC21kVapqwCLRDgwMEA19AOBADweD2ZmZhCPxxGNRiGVSnmxEuQLfDUSdHZ2oq2tDdu2baPzAdy5CJIGK/e5t7qBSr2uWQnOatKvJr2TL/1rMplyIky73d7SKQVu+qaUvj4fU718IJPJ5EzK9vb2Ynh4uClfjGZLK3M1c8bHx3Hs2DGIxWLaEaJQKOhsQLNkIuoR/RKZbW49gIilcbthiqXBCM4Ew5N6ag2Vg7Oa9Mvt0yepBIfDQfP0a9asKThw0ypFWC6KpW/qrX1TC0hxMx6P48CBAzAYDEVf82Ko1/pb5QQALBaEu7q6YDAYqFaMx+PJkUsgXUGNShM06vXJF0vLT4MV6og6EyL9fJytLZtNwXKRPjdP7/f7odVqMTg4WDJP30qkn81mMTk5WfXwVDNIP5lM0klZpVIJiUSC7du3nxG1kUpQD2llrlYMkU8m7ZFTU1NUJkKn0+VMH/ONZkWq3DRYfkcUkUjg2z6x2amYeuCcI/14PE776RUKBUwmE8bGxsp+Y5tN+tz0TTweR0dHR9XDU40ifZZladteIpGAyWTCpk2bIJVKsW/fvqrJqdb1t1JEXw3y2yOTySQ8Hg+VDe7o6KCbQDWTssXQCq9bfkdUJpOB3++HxWKBz+dDOBzOsZKslrgboQAqRPo8QqFQIJFIIBaLwe12Y2FhAceOHYPRaKy6E6QZ0XGx9M3hw4fR3d3dNNIshfyi7NDQUMsUwM+kk0W5a21ra8tx0iKTsq+//jpSqRRvNoLNzkkXglgshl6vB8uyUCqVGBwchM/ng8ViQSAQoLr5pBZS7mvais+1Vpy1pE9s06anp/GOd7wDP/3pT6FQKLB58+aarstnpF8q/1iq+4YUYltJ8IxblGUYBr29vXU1UakF2WwWs7OzcLlctIjYah1NtXQncSdluTaCJ0+epCYqer2+4oG8Vnp98kG+U9zpWK5u/okTJxCNRtHZ2UlPScvVvoT0zhmCX//61/jXf/1XXHjhhWhvb8ezzz5LLRRrBd8Szflftkq6b2qVV+aL9El9xG63IxAIQK/XY/Xq1bymFAqhmvWzLEs7QsLhcM4UNRHHe/XVV2mRsJiC5pkGkUgEqVRKbQSJiQqRCCAkqNfrSzYANEv/vhwUIul83fxsNotgMAiv10snpLnzAdzvWj36/puNVnLO+gyAj2JRhsE9MzODwcFBAItHt/Xr1wMABgYG8OSTTy57r7e97W248sorkclk8Ja3vIUSdauYqAC5hNWs7ptaH59KpZBMJrF//34oFAr09vZW7AvQKCQSCTrVS05NmUwGfX19VEffZDLB4/HAbDZDLBZTBc10Og2tVguDwQCNRlP2PVtJWjn/OvkmKsVkIjQazZKAo5VJv5x0DJFM1mg0dEKa+CacPHmSqmXq9fol4mh8rK/Zr10rOWcdBLCNZdkowzC3fu5zn7t09+7dABZz85VE6eSLKRKJkEwm6X/zAb6iY5FIRHOuzeq+qSVSttvttB1248aNaGtrq3od1aLU+kkB2WazIZVKoaenh3oYLGfKXig1Qlp6jx07hq3hMKRSKZKxWNHTTCtEdPko9rlaTiaC6OeTU097e3tLEFcxVFN4lUgkOWJ5xDdhbm4OHo8HMpkMDMNAr9dDoVDU9NyLnUQaiZZxzmJZ9s+cv99jsVhqvikf7lKFrllLpE/SN6FQCFNTUzXlvGtdSyWkH4vFYLfb4Xa7odFoMDAwAJVKhQMHDrScNEI8HofNZoPH44FGoyk41VvJF40rm8CyLMRKJVLJJJUSJn30lRhpVAK+Iv1yUUgmgtQCIpEIlEolPeU1Y7NfDnwUXolvQm9vL6anp6m16cTEBGKxGNRqNa0HVPr8m62lD/BI+gzDiFmWzRT5dbnOWQQfueKKK+g/4vE4tm3bBolEgttuuw1XX311zeutFtUQbaH0jUqlwurVq2vKF9c70s9kMnC73bDbFxWve3p6sG3btpwPbavIGRADFZvNhkwmg97e3rrYODIMA7FIBLFcjm3btlEdfWKkIZPJaFqAD9Sj379ScPVystksbDYb5ubmcPDgQbAsW1I6uZGoR4slSV2S2Qgi5z03N4dsNptjJVnq88Z3uqgaVET6DMP8K4CHWZadLvDrzwD4Zq0LYhjmBgDbPvvZz9Kfzc7Ooq+vD1NTU7j00kuxfv16jI6Oln1NPo+jlZD+ct03oVCo7s5X1T4+FArBZrNhYWEBBoMBq1atWrYo2yzSZxgGyWQSk5OT8Hg80Gq1GBsbQ/tpZ6tGIF9Hn0yNxmIx7Nmzh+bGm2WszgUf3wGRSISOjg6o1WqsXbuWSifbbDYcO3asYpkIvj872WyWVyP3/HQMVyZjbGyMpsLIpk+6onQ6HTo7O5e85q3QDVTpp/ATAP6BYZjPAXiKZdkg53cfYRjmeyzLFjIfLemcBQAMw1wO4HYAb5XJZE7y876+PgDAyMgILrnkEhw8eLAs0mcYpi4aIct9UPO7b3p6egqmb/jQvan1Gtznkkql6KQsOd6WU5RtRm43m83C7XZjdnYW6XQao6OjTdPqyQeZGrXb7di8eTNCoRAVEJNIJNDr9TAYDEW1Y/LBZyGXL3DXxJVOXk4mopiLFt/1gUbLMOSnwkhX1OzsLB2QI5uAUqk8I0l/EsAtWCzKXsEwzN2cYmwAQBuAQqRfjnPWZgD/AeDvWJZ1kZ/7/X4olUrIZDJ4PB68+OKL+NznPlfhsvlDoUi/mu4bPtIifNQs0uk0Xn/9dcRiMZhMJmzYsKGiPGUj0ztcW0S9Xg+z2YxoNEq9WytFvdedPzFL9NNPnTqFSCQCtVpNo+J6nwL4JMNi1yokE0FUM4lMBOmKITIRfA8/NdtEJb8rKhwOw+v10k2Q1ENSqRTdBM+EQq6VZdmLGYb5EoBHGIb5vwAeBpDCIukvAVuec9Y3AXQAeIxhGLzrXe/Ck08+iWPHjuGWW26hZHvbbbdhzZo1FS2Y+yGt9cPPJblajMP5aP2slnC51o7pdJoWZat5XepN+qSuYLPZIBaL0dvbS20RfT4fotFo3e7NN+RyOcxmM8xm8xIJZdIdYjAYlrwXfJFCvUk/H0QamfgCJJNJeL1ezM/P4/XXX0dHRwe6urp4lTVpJRMVbhfY0NAQstks5ufnYbVa8corr4BlWZjN5oY7v1VK+hkAcgBgWfarDMM8DuAhAGsBjAAoGtqypZ2zLs9/CABcdNFFeO211ypc5htoa2tDKpWCTCajRFtLcS+TySCZTNKulWLpm1LgI0qvhHAzmQw8Hg/sdjuy2SxtX3zllVfQ2dnZkDVUgnA4DJvNBr/fD4PBwPuw13Lti/W+B7BUQpm0CXKHpQwGA28y3vVK71SCtra2nCnZcDgMp9OJaDSKv/3tb+jq6qIWktWefOpxcuCzHkh8cletWoVUKkW1wRqJSl/ZBAAJs/gqSFiWPQbgPIZhvozFCL7lZu2JOXotpJ+fvgFQ1vDUcuAjp1/OaYFMyvr9fuj1eqxYsYJXHXY+SJ98sTKZDJxOJ+x2O6RSKXp7e5cVw6vXhtOMwjS3TZAMS7ndbni9XoTDYRiNxpy0SKVoRHqnEpAouK2tDQsLC9i8efOSASmS+qrkFNpKkX4hZDIZyj9SqbQp7ZuVkv59LMtOMwxzP4BvATjFMIyEZdl/ZRjmRwBcJR7fcLS1tSEej1PnnUq+0MXSN/v376+J8IH6RvqpVIoWk4kGSSVKonysoRJwJRwMBkPNG2oroNauKjIsFYvF0NfXh2QySdMiKpWKEmK5r1OrkT73WiKRiAqm6fV6AEtlIsp9zq1uosK9XrO63iol/d+e/v8xAJrT/51hGEbEsux84Yc0FyTSBxaJtlT/dLndN7WCr+6dYlIORqOx4qJstajmw0te52g0ipmZGfT19VUl4dAKMwL1hkQigVarpR0y4XAYbrebSiaQ4mij+uT5JP1ikXl+QZR0QXFlIgr1xje6e4eP67V6IZfBYq59H4AoALCL37qW/eaVY5lYrfZNLeAjQmYYBvF4HNPT03C73VCpVDCbzQX7g+uFSu5DvrxWqxWhUAhGoxEKhQLr1q2r6otV63Os9vVnDh+u6b6VIJ/EuMXBkZERpNNp6lVw7NgxKJVKGhFzayCtHOmX0xbc2dmJzs5O+py5vfFcmYh6aOXUk/Sb0fJcEemzLJtlGEYF4F4AYQBgGEYJILHMNG5TIZPJKOnnE20t3Te1opbunWw2C4/HA6fTCYZhMDg4SJUiG41yNi9uuolMN65atQoMw8Dj8TQlWm9V7ZhKIZFIYDQaYTQawbIsIpFIjoY+EYqTSqUNK1JXgmo2kPze+FgsRmsBXq+X6ixVI5OQj1ZPF1WDSidy3wrgPgBrAHyTYZhvA3gfgD0AGhf+VABupC8Wi5FMJmG1WmtO39Qa7VRD+uFwGHa7HT6fL8cVqNo+dT5QjPRZlkUgEIDNZkMkEimabqrlxMPHaSkej9PCMZmqPVPBMAw6OjqomxqRiHA6nfB6vUin05ibm6vZVJ3PYikfpwaFQoH+/n709/fj0KFD6O7uRiQSoTIJtaS/6lEY5nNiuBpUGhp+HMAPWJb9CcMwvwYwCmAci1H/4dO5/dYwkD0NuVyOUCiEubk5+P1++Hw+9Pb21pS+IWTDV7//cuDWGEhHy+joKEQiEebm5pqe085/HqlUCna7HU6nE+3t7ejt7a2rX2s1YE+rb0ajURw9ehS9vb0Qi8XUbziRSMBsNtcsncDXJG211+FKRIRCIZw8eRIAcOzYMSQSCdoima8hXw74LuTyCY1GQzu/UqkUVYUlMhGVOGid85E+FoXSyDYVBDAEQFTFdRqC48eP4+DBg/jtb3+LW2+9FZdccgm6u7tph0C1qPc0LcuyWFhYgN1up+16hTapZvv1EhCDbpvNhmg0ip6eHuqBWwqNjPSTySRsNhtcLhc6Ozshk8mwZcsWMMyinn5fXx+OHDmCjo4O+P1+nDp1ip4AiHRCuWj2ZlwIUqmUGouT98ztdlMnLYPBQOWTlyPDVh+m4l5PKpXmpL+ITMTx48cRi8VKykTUs2WzWaiUrF8AcBvDMP0A1ABuxqJ65onTv2+JTzrLsrjyyisBACaTCbfccguuueYaTE1N8XJ9Poa8SF86F1yjD5VKVTJKbqQEQiEkk0mEw2EcPXoUGo2moTWRckE2UKvVing8jt7eXmzZsgUikQgHDhxY8vekaEgMfEi+mJCEVqvFhmwWTIOiNT61d7jXyZeI4ObFo9EoJUOdTlfwtMMXETZSe6damYh6dgO1fCEXi1H+cSzKJZwEYAewi2VZWzmpHaa0e5YMi5IOW8877zzs3r0bQ0NDAIC7774bP/7xjyEWi/Hd734XO3bsWO4+2L17Nzo6OvDVr36VEiNfkTGfEgqkKGu325FOp9HT01N2UZaPtk+gsi8eyy6aqNhsNiQSCYjFYoyOjtJR+0pR68ZV7LEkzeRwOKBSqTAwMJAzeVzuPbn54mw2C5/PB5ZlkUmnceDAAXoKqLc1ZK0o9XzznyeXDEkPvcFgQEdHR0tM9xZDJZF5KZmI9vZ2JJNJxONx3t7fMzG98wSAX3E7dRiG6WAYxsiyrLP4w8p2z/oIAD/LsmO7du1iP//5z2P37t04evQodu3ahSNHjsBms+Hyyy/HiRMnlo20iWmGTCajffpisZgXnXM+SD+ZTFKtep1OV5UkMF9tn+V88RKJBGw2G9xuN9RqNYaGhqBSqXDixImaPsS1pne4IJOsVqsVkUgEPT092Lx5M2+FM5FIBL1eD5FYDBGAVatWwePxUDMVEiV2dXXxcj++US655pNhIpGgG0AoFIJEIkFHR0eOaFi1aCXBtUIyEfv376fvLzcVVG2t50wk/U8A2M4s+tk6AXQC+AqA1wE8UCLaL+medfrfdwHAzp078fGPfxwsy+I3v/kNrr/+eshkMgwPD2NsbAx79+7FhRdeWHLB3O4dviLjaq/DLcpms1nI5XJs3ry56g9BvUXbSMGzkN1gOY9vFNLpNBwOB+x2O5RKJfr6+hpSPOamCjKZDDVWJw5LVqsVBoOhJqOceqV3KoFMJssZlDp+/Dii0SgOHDgAhmGg0+lgMBiqmg+pd06/WpB5CJlMRs1y/H4/vF4vJicn6emnmG7+cus709I7/xfA/wHwOID/BrAJi5LKjwKLffzLPLYc9yz6NxKJBGq1Gl6vF1arFRdccAH9I7PZDKt1iRx/QSgUCvj9fgD8pnfKJTpu62I4HKaDX+FwGAsLC02Lkpe7Rjl2g3ytoZbHR6NRLCws4JVXXoHJZGqaVy+weIrkWiq++OKLSKVSOHz4MDKZDB0e0mg0Tfmi87UxMwwDuVwOlUpF5SG4+vEqlQoGgwE6na6s7rh65PTrEUkXk4mYmZnJkYnQ6XTLbvJnnAzDaVL/BsMwCQDfAfAbAF9nWXahDmvjBfkTuXyoFpazeXCLsh0dHejp6cn5wkej0aZJKxe6Ri12g40kfa4om0gkglwux6ZNmyomjuWKfbWCYRiIxWIMDQ3RCVKPxwOLxYIjR45QYtTr9WVtUs2O9Je7Vn5KJBQK5UhEEKIk2leFrtWqqpjLoZhMxOHDh5FOp6ljWr5MxBkX6TMM04VFobU+LMopXwHglwzD3Mmy7O+Y5X1yy3HPIn9jSafTCAQC0Ol06Ovrw/z8G4cEi8VC3bRKIX84q545fUKcdrsdqVQKJpOpaFGWL8KudePIZrOYnp6G3++vym6wUR9aIrW8sLBAT0upVAqzs7NVr6HekRZZl0QiyXGXIsR48OBBAKDEWChN0OzUWSEUI1auXMLo6Ci1UrRarTh69Cja29vpcyXRcKNIup4oJBNB2mGJTASp95xxHrkALgQQY1n2707/+xjDMH8EsA0ASkgxlHTPAvAkgBsBvPT444/j0ksvBcMwuOqqq/D+978fn/nMZ2Cz2XDy5Emcd955ZS2YK7jGV/45n2y5jk5arXbZdAgBHyqb1V6D2A2SvnqTyVS1Cmc90zvZbBYul4saqPT19WF8fJySBF9a8/ng4zNS7Br5xJhMJuHxeGh6RK1W0/QIn85KzdDeybdSDIfD8Hg8eO2112g0zDBM072D+YZEIsnxTSbtsJOTk/D7/Th+/Di6u7tr8g2oaX2V/DHLsk8zDPMMs6i/I8GiaYoLwM8YhlEA0LEsayny2HLcs358+lqntm/fjl27dgEA1q5di/e+971Ys2YNJBIJ/v3f/73sHvlyBNcqBUkT2Ww22O12KudAHJ3KAR9ReqWEG4lEYLPZ4PP5oNfrsXLlSpw6dQo6na4mdyC+o9FoNAqbzQav11sXAxWCVogw29racnT0A4EA3G43pqenIRaLkUgkEIlEIJPJalpvswXXSGFUpVJheHiYiqbNzMwgEolgYWGhIkP1RoCv14zbDvvSSy+hv78fPp8Pc3Nz2L59e8NlGSpN76zFotiaG4s9+1ksSiw/h8Xi7rVY7MMvCLa0e1YcwHXkn9y/u/3223H77bdXslwAuZG+WCyuiWhJS6DX64XT6URvby/Wrl1bVXdGo+wSC9kNEhmHcq9R6xrKeTyZV7DZbGBZFn19fSU30VboHOITDMNAo9FAo9FQN639+/djZmYGExMT0Gg0MBgMNctD1Ao+yJCIpiWTSZr7J4bqtUpE8AW+O4uAxdeuq6uLZgRaPtIHYMNiNB4BkMaivLIIi+2bXgBP8bo6HpCvsllNTj+ZTNKirFKpRGdnJzQaDUwmU9Xr4oOwlts4uDlwvV5fNFrmYx21PJ74hpL6Dd/OXsuh1TcMmUwGuVyOdevWQSqVYmFhAW63G6dOnaJywuXKQzQ70l/uWiKRaEn7K1cigvtcy9HL4Qv16qnnrr/lC7lYFFY7AMDCLsosj2JxOtfCsmwCwCTfC6wV+SYq5X7RyfSlzWZDMpnM0ZPhQ+isHpF+IbtBbg68nGtUimrqCtz+/0AggP7+foyPjzctomtl8udOk3MHpmKxGNxud448BDkFFOuSaUXSLyRnkt8emf9cuaeAekbKrTBIVQ9U+oqtAPAfLMu+iWGYrQCeweJw1f8AuKNE905ToFAokEwmAZRHtNx8cldXV8GibKNNzUtdIxQKUQKt1ACmkZE+t42V9P/Pz89Dp9NVRfh8pJbOBBRap0KhoOJp3Mj4+PHjUCgUNDKuRy2k0RsI97kSoThSGOV66ZZqnqgUjSD9MyHST2IxrQMAbwLwdQA/BPAygDvQIoJrXJST089kMnC5XLT3u7e3d9l8Mh9Req3XSKfTcLlc8Hg8tK++GrvBehMny7JUgZMInpXb/98IsKcF2aRSKTQaDf0ZH+Cr66YU8iPjSCQCt9tNjVSq3VSXW1OzTg35QnHxeJxuAOFwGIlEAg6HI6f7qZa1CZE+EAcgZhhmBYA3A/hXACYA/tO/b0nSLyTDQIqydrsdwWAQBoMBa9asKasoy8eQVzVkS9Zss9kQCoWg1WqhVquxfv36qtdR66ml2PMoJXhW6vHlotrHptNpWijt6OiARCLB0aNHwbIsxGIxVCpV06Z7awXJjw8NDdEumdnZWYRCIUQiEXoKqNZPoh45/Wohl8thNpthNpuRyWTw4osvIhQKYWZmBgzDLDsDUQpCemcRNgA/B/ASgEdPt1z2AXgEoH65LQWFQpGT0yfuQU6nE0qlEj09PVi5cmVFHwg+2i0rifSL2Q0mEgmcOHGi9AWWAZ/dO2RT4gqelVIMreX+1RBPOByGxWJBMBgEAFqnaWtrA8uyeO2115BIJPDKK6+AYRjab93R0dG0dFAt9yVdMtlsFpFIBN3d3TkG42QDqESrqFXrAwyz6IswPj6O8fHxJTMQnZ2ddBMoZ0OvR/dOK6DSPv0MgP84/T/yMyuAB3heF28g3TtPPfUUBgYGEIlEIBKJyjb5KIRG5PS5mj2RSIRqy3DX3ApTvQzDIJVKwWKxwG63o729HX19fQ0xZy/3+mQYzWq1QiKRoK+vDytXrsS+ffsgEomQSqWodEJbWxsdjiLqkqdOnUI0GkVXVxcMBgPKmwXnB3zFUYRcub3yZGKWSAmXS4p8F3L5vBY3Ms+fgQgGg/B4PDh48CBYlqVCccU2PL4j/VaJiasqfTOnXyES2TMMw7RilD85OYmf/OQnmJubwxNPPIF77rmHWuHVAr46bwqhErvBRvX6FwNJj8XjcfT391e1kdYzvUOE49xud07bajabRTqdRjqdht1uh16vpxuDx+NBd3c3MpkMJY2+vj5aQHS5XDCm02AYBhaLJUdSoF6o10Ru/sQsIUVyyiGngHxjnFaN9JdLFTEMA7VaDbVanSMRQTa8jo4OqodE0l6NsEo8Ewq5AJamcVqR8H//+9/jvvvuw80334yenh786Ec/AgBMT0/XfG2+bQpJMZG4O5lMprIIlM8OoHLBbQuVyWRUNmBgYKAh9y8FUji2Wq1IJpPo6+vD9u3b6XuWSqUALL6HGzduhNPpxCuvvIJkMgmVSoXR0VGoVCpks1kahTIMQ1smiZMUy7I5KpokDcT3CYfP16ZU6y6XFElqZHp6GqFQiGrJ6/X6lsrpc1HJqaGYRARJe2m1WrS1tfH6XrZKjaBpY30Mw2gB7Maiz+4MgPeyLOvn/s2hQ4dw6623IhgMQiwW4/bbb8f73vc+AMBNN92EF154AWq1GgDw4IMPYtOmTfSxO3bswI4dO8CyLO65J8egq2bwkd4BFj8Es7OzcDqdtNhZid1gI0m/kOCZTCaDw+GgLbGNBnftqVQKDocDDocDHR0d1OQlm82CZVkkk0lK3hKJBKFQCPPz84hEIhgaGoJOp8PCwgIcDgcmJydpL3hXV1dOCow7yTw8PEzTJNzcsUajQSqVQjqdbhldmUo/J/mpEeKkNT09jVgshra2NpjN5pJ+uqXAZ3qn2g2kmETE3NwcgsEgYrEY3fBqaYE950kfwG0A/siy7D0Mw9x2+t+f5/6BUqnEww8/jPHxcdhsNmzduhU7duygrXXf/OY3sXPnzmVvQj5QfBeMqo30WfYNu8FYLAaJRFK2PSKf6+BeoxghcCUcJBJJwWGvZss4pNNpTExMIBgM5gzQkaieEAF5fUluv62tDf39/ZTUgcXPW29vb04v+MmTJ6FUKumXXiqVAqfXmz6d5iEqmkReeGFhAS6XC3v37qU1gu7u7qoJo17pnUru39XVha6uLoyPj+PAgQOQSqXUT5dskNW0hrZifYAUv8kMjMlkynFIKyabXM76znXSfzeAS07/90MAnkce6a9YsYL+d29vL7q7u+F2uynplwu+82bVpHcK2Q3G4/GyJaILgY/nVYh08wXPlmtlbYb+DVHftFgsiMViGBkZwcqVK8Gy7JKoXiwWI5lMYm5ujubr169fv2wentsLzrIsIpEIVYdkWRZvOm2MTu6XnwbSaDSQy+W46KKLEI1G4Xa7cywVu7u7yzZTaVR6pxKIxWL09PRgdHQ0Z4Mk8hAk1VWOnAbfpF+PHHxHRwc9PWYyGfh8PiqbLJPJaEBQSg5DIH3AyLKs/fR/OwAYl/vjvXv3IplMYnR0lP7s9ttvx1e+8hVcdtlluOeee6ruO64U5ZI+V8ohlUrxPpjEJ+kTwTOr1QqGYUoOqOU/vtb7l4N4PA6r1QqPx5OjEqrRaJBOpymBSCQSiEQiLCwsYH5+HqlUCv39/TlCc5Wsj/ulT6VSwOkT1ssvv0xz3V1dXfRzEYvFkEqlkM1moVAoMDg4iMHBQaTTaaovf+TIEXR2dtLi4XL1m0YNeVVzrfxhqWg0ukQ4rZQ8BF9EyOcGAhQmabFYnCObTJ5vORIRZ3Qht1wwDPMcFoe38pEjl8myLMswTNFPpd1uxwc/+EE89NBD9EW7++67YTKZkEwm8bGPfQz33nsv7rzzzoKPF4lEyGQy9E2o9cNRiqhisRjsdntZdoPNBvHtnZmZgU6nw8qVKysWPKsn6ZPCrMViQTqdRm9vL4aHhyESiZBIJJBKpZBIJNDW1kZNcux2O2w2G9WsV6lUVa8vH1KplH4Gzz///BwRNHJ/sViMFStWUHE/cgoglopGo5F2y7jdbszMzOSQCfezwnfLJl/XKkbUSqUyRx6CRMVEHoI8R3LSqmfLZiOux32+xSQiiCheIZ2hZqCupM+y7OXFfscwjJNhmB6WZe0Mw/RgUZd/CYLBIN75znfi61//eo5Pbk9PD4DFPvybb74Z3/rWt4qug0zlSiQSSjK1fNAKRfokUrbb7chmswVNxFsF7GnBM6vVSnOy69atq7oIVg+QwqzdbkdnZyeGh4cLFmb1ej2OHDlCiTWVSqGvrw9bt26tu065SCRCZ2cnIpEI/H4/ZDIZFAoFnQgNhULQ6/VQqVQF00CdnZ1Qq9UYGxujcgInTpygAmokr8wX+CT9cpAfFRN5CK6JCpmW52tdfEf6lXx/i0lEkNoH2eiaXeBvZnqHuGTdc/r/f5P/B8lkEu95z3vwoQ99aEnB1m6308LZE088gXXr1hW9kUwmQzweR3t7OyXsWiICLulHo1Ea1et0uortBvn+oC6HfMGz0dFRas7eLBOV/MeHQiFYrVZamN28eXPRwizpuggEAgAW5QcikQicTifS6XTBHnO+wLIsjh8/Dp/PRzd47iZD+sDn5uYQDodpays3DQQsfpYYhoFMJsuRE/D5fHA6nQiHw3j11VfR3d2d00NezXr5QrUpmULyEA6HA4cOHcrxDa7lOfI9TMWXRATp1HM4HNi3bx/EYjE9ITQazST9ewA8yjDMRwDMAngvADAMsw3AP7Isi0cffRR/+ctf4PV68eCDDwJ4ozXzAx/4ANxuN1iWxaZNm3D//fcXvREhfYC/Hvt4PE49Tvv6+mjKoRLwceoohfy+9fwTSCAQaGr3DbAYUTkcDtolZDabCxZmyZg9kdJwuVzQ6XRYu3ZtTmcMl3C5Pea1GnKQzque07pLXV1dRUXuuH3g2WyWOmJNTk5CJpPRqV8uwXFPASQt4Pf7MTY2BrfbjUOHDoFlWRo9V7Kh8Z3eqRWkQ8ZqtWLFihV0OI48x1LTsoXAt2wCn+ki4hmg1+sxPj6OeDxeN7vPUmga6bMs6wVwWYGf7wfwUQAfueGGG3DDDTcUfPyf/vSnsu/Fl2Ui124wk8lUlf/mgo9TRzEkk0nat97Z2Un71vPBhwxDtSQQj8epAqfJZKJdQtlsFplMhn6JSWE2GAxifn4e8XgcZrMZ5513XkESzydcknefnJykeeVy9VcA0Mldq9WKzs5O9InFYBgG3d3dZT1eJBLRlkcAtKPn6NGjyGQy1DSbmwYijyOFZJVKhZGRETo0NTU1hUgkAo1Gg+7u7rI2tFadohWLxWhvb6fPsZA8RDkF73rk9Ou1ichkMt6Hv8pFa0yO1Bn5RiqVkFwxu8EDBw7U7PBUj2nUQCBAc/X1Fjyr5vH5hdm2tjb09PSgp6cnZ2KWRPUsy8LpdMJqtVJnpUqmXrnmIyzLUsJ99dVXASCn0JZ/zUgkAovFsiSFU+sXValU0o6eVCoFn8+H+fl5hEIhqNVq2ipqtVohl8vpPAB5TcjQFNnQXC4XTp48CZlMhu7u7pxCKfd15wv17q0vJA/hdrsxOzvbMHkIsrZ6FoabQfjAOUT6lUb6+XaD5couVwK+Uk3pdJoWPdvb22E2m8smxkaRfrHC7MzMDP09AJqvj8VimJmZgc/ng8lkwubNm2uWOmYYJievTKLmycnJnHa7bDYLi8WCbDaL/v7+qnwKyoVUKoXRaITRaEQqlcLU1BQmJiYALG4OpNgpk8kKFoOJ3yqwtFDKVdBsZb2ccuUhxsbGcuQhuLUSnU7XlO6dZl6vWpwTpM/1yV2OaKuxG6wFtRJuMBiktQWj0dgUwbNSCIVCsFgsCIfDlLy5hdm2tjbMzc0hkUjAYDAgkUhgfn4eAGA2mzE+Pl63LwpXaiCRSGBqaooOYKnVavT19eVM7NYL0WgU8/Pz8Pl86O3txUUXXQSpVEptAo8dO0bNUIg2fKE0kFKpxNDQEC2UejwemiIhJx6DwVBz50gztXe47xm3VjI1NYVMJgOFQoFwOFyzPARQH9JvhW6+c4L0S6V3arEbrAXVRPpkY7LZbJDL5ZBIJNi6dWvVH856mKiQiVkid9DX14dVq1aBYRaN6bmF2d7eXmi1Wpw6dQqzs7MQiUSL8sV9fRUV8apFJBLB/Pw8FhYW0NPTg4svvphq85CUgkQiocVTvuwHSZprbm4O6XQaAwMDS04UXJtA7mDXsWPHqAwyifIJuDMBRqORpkiOHj2KWCyGffv20efT3d1dVYqyVU4N+bWSmZkZLCws5MhDkMGwasiW724g7qxQM9H8FTQAhUifDCURtci+vr66HuMLoZIoOxwOw2q10o1p/fr1kMlk2L9/f8PWUOrx3IlZg8GAtWvX5hRmyd9JJBKIxeIc0bPe3l6sXbsWDMPA5/PlkBs5vvMVJbEsS6NgAOjv719ipNPZ2UmHu+LxeE7E/ZZMhsowVPp5yWQycDgcsFgsaG9vx8jISEFHsXxIJBKaBiK1G4/Hg5mZGUil0hxBsEIKoUQuwGg05pwgEokE7ZTRaDRlk1yr6eUAi6+RVqvNGZRyu9203kHSXeVudEJ65wwGyemzp+VwZ2dnkUqlYDQasWHDhqbZ4pWK9PMFzwptTI0uxBZCPB7H4cOHkU6nc9pXs9lsTlRPJImXEz0DQKNqQm7k+C6Xy2m/ejXvGfEqsNlsUKvVWLFiRVmT0nK5HP39/ejv719sszvd8bRnzx6aUy4VTZK0ldvtRnd3NzZt2lT1aZJhGGg0Gmg0GoyNjSEWi8Hj8WBiYgKpVIoKgpFcfjgchtfrRWdnJ7LZLORyec7UrNfrhd1ux7Fjx3J05RvxvahXUbiUPARJdZG5iVLX42t9QiG3gXj22Wdx+PBh7NixA3q9HsPDw0170QmKpVai0SisVit8Pl/dBc+qfTyXQFmWxbp163ImZklhViwWQyKRIJFIVCR6RtZGyG18fByRSAQulwuvvvoq7eIoJ0XBTeEQ/aNqp3VJ+ygAXHDBBTntoHK5fMmAUTAYxNzcHKLRKMxmM84//3zeoz2FQpGzKZGT0uuvvw6WZSGVSjE8PAyNRrNEGoKk07iqkm63O8cusru7m5cceTHwSfrF3tdC8hBOpxMTExNURTW/60mI9M9AvPjii/iP//gP/PnPf8ab3/xmfOxjH6M6LXx80PjQ8CGRfr7gWV9fX1kCYY3us88vzK5evRrz8/Nob2+nRA8skj1fomcE7e3tVMM+kUhQTReSouju7qZdS+WkcGoFwzA5OWXSPXP48GEkk0lks1kolUoMDw83pCAMvKEzRXLaOp0OkUgEc3NzsFqtNFWmVCoLpoFUKhVNaxG7SG6OnG9pCD5R7veRKw+Rr6KaTqdpuiuTyQikzzeYMoxUgMU3af369QCAgYEBPPnkkwAWXbCuv/56eL1ebN26FT/72c9yjqT79+/HLbfcgvPPPx/xeBxGo5G24vGwdl5IP5FIYHJyEl6vF1qttuKBr3oUYvOxXGE2HA4jmUwimUxCKpU2RPQMwBLZAjLMEwwGIZFIqIzxypUrK5LFqAXks0d0ZRQKBYLBII4fP56jwV6PLz7xKXY4HDAYDNi4ceOS9BFX34dslDqdjhbM87uByHudbxcZiURw8OBBSpyNUrcthWoKr2T4jaiocgvmfr8fR48ehdFohMFgqDndJaR3FlHSSAVYPL4eOnRoyYM///nP49Of/jSuv/56/OM//iN+/OMf49Zbb6W//+QnPwkAOHHiBNVn4as3vpZpWiJ45vF44PP5MDg4WJWMA1Df9E4sFoPNZlu2MCuVStHZ2YnXXnuNdiZwZaTrLXoGLAYFSqWSni5IS2MgEMCpU6doHaBeayGRdCAQoPaM3C4NblHxxIkTaG9vL2vCtNJ7LzelDORqwZAUh8PhwPHjx2kuP18SmHsKIHaRPp8P4+PjdMgtm83S9AjfdpGVgI8cPLdgHolEMDw8DL/fT83Ua3meQsvmIkoaqRQDy7L405/+hF/84hcAgBtvvBF33XVXDukT5A9n8aF5UU1ahWukotFo6DCQXq9v6DryH88lfaIvY7VayyrMAoBarUYoFAIAuil4PB6aE+arzTEfpCg8Pz8PkUiE/v5+egIhvyd1gIMHD+Yc62te0+mNe3Z2FgCW3JuLfFOWcDgMt9tN11RpVwm33TOTyWBgYKDovYshP8VBcvlzc3N0TXq9vmAaiAy5dXR0FLWLJNIQjWxRrIfgmlqtRldXF5WH4D7PcuUhCLjpokYKLeaj2aRflpFKPB7Htm3bIJFIcNttt+Hqq6+G1+uFRqOhHyqz2Qyr1VrwJuUOZ1WCctMq+YJnXCOVmZmZmtfCR3qHDErZ7XY4HA6o1WrqAVCsMEu6oIqJnnHbHMl0KF8FwVQqBavVCofDAY1Gg1WrVhVM4XCP7iMjI0taL0kdoBLhMqIJlM1m4XQ6y+4A4q6J+LGSNRETDm77ZKEZBXLP+fl5KJVK3tJmDMMsaVElufx4PE4j/FQqhfn5eahUKqTTaZoGkkgkOVIagUAALpeLOmkRaYh6bf4E9ei24V5PKpXS55kvD8EVyevo6Ci4jnMmp8/wYKQyOzuLvr4+TE1N4dJLL8X69eupIXo5UCgUvKtslrpOOYJnfKyl1vRONBrFwsICDh06lKPVw9XBAd4ozJYresZtcyQREpE7IFrxlQ5fhcNhalbd29tLA4Fykd96SSL1cDhMo9NiOfd4PI75+Xl4PB5cwrKQiMVYs2ZN2fdebk35tYn8GYXOzk66IRsMBmzYsIF3SZBia0omk5icnMRrr70GADTyzdeE56aByCkWwBK7SEKM5dpFVoJ6kGqxNebLQ5Ci9+TkJCKRSI48BHmdzhnS58NIhfjIjoyM4JJLLsHBgwdx7bXXYmFhgX74LBZLUb9ZvlQ2uSh0Ha7gWSwWKyl4xkePfDXXyGQycLlcsNlsEIlEkMvl2LRpE52Y5ergiMViGmFWK3rGjZBILpkQm1qtpqmAYnZ6+Smc1atX85q75QqXkZw7WVM0GqUyEbT7qE55WeKsRTpknE4nVdNUKBQwm83o7u6uK+ETJJNJzM/Pw+VywWg00kllkpo6dOgQjW5JGqiQNlApu8hUKoVUKsVLvaWZKRMy4EmK3tz5EjIBnUqlmt4mDjQ/vVPSSMXv90OpVEImk8Hj8eDFF1/E5z73OTAMg7e97W14/PHHcf311+Ohhx7Cu9/97oI3qUVlsxi4uXQiJuZwOCoSPKs1NUOuUe7zicVisFqt1PB87dq1yGazmJyczJmY5YqeTU9P8yp6lp9L5qpEErIlAzVWqxV2ux1arRarV6+uWdW0GPKVOIPBIGZmZnD06FFqAr5ixYqGkC15TWZnZ5HJZDA6Ogq9Xk+naI8cOYJMJlMylVAtuIXh/v7+JSc5bmqKG90SwTpuaop8LvOlIbh2kR6PB/v376efC5ICrAatEknny0OQtGI0GsW+fftoLc9oXNYWvG5oNukXNVL5yEc+gh/96Ec4duwYbrnlFkput912Gz1W33vvvbj++utxxx13YPPmzfjIRz5S8Cb1yumHw+GcnvVKBc9qLcKSayy3cZDCrMViQSaTQV9fHzU8z2aziEQiCIfD1PlLKpVSqV+WZdHf31830TNunzspcFosFkxMTNCOkI0bN9aN7PORTCZhsVjgdDqh1+ux9a67gEgEUz/6ESVbnU6HtSwLvuO1Uvn6fDlmojQZiUQo2dbSDko2GqIDVE5hmBvdZjIZ2tJ5/PhxahhCPlOFTgGdnZ2Qy+W48MILKTESg3GSAqzkOfFdyOULJK04Pz+f46tsMhXKetcfTSX9EkYqAICLLrqI5hPzMTIygr1795a8D5+RPtHs8Xg8iMViGBoaqjo/Wc/0Tn5hdnR0tGBhVqlUYu3atXA6nTh16hSy2Swt6Gk0mprWVi6Ia5LFYqEzGQqFghqNcN2i6tFzHwqFqMsWt04hmZgAY7FgwGzGwMAAJVvyGk5MTKC7u7sizZp85PfXl5Ov56bLSDso6b8n7aA6na7kqYykzmZnZyGTyTA0NFRRrYwLbscP2cA9Hg8OHz4MAPR37e3tdAMIh8P0v2UyGa235E/MkudUqlee70JuPSAWi6HT6Ro2rFcIzY70GwKFQoFkMgmgetLPFzwjH0JyhKsG9SjkBoNBWK1WhMPhgoVZkvcUi8VU9MxqtSISiVDxL6/Xi+PHj0MqldIccz10WJLJJO3CKZTCIZFtMpmkPe6JRIKXnnAysTs3NweRSISBgQGsWbMm53qZSy6B9OGHIXriCWSvuYaSrVgsBk5vRE6nE8ePH6cer9zC3XIgtYKFhQX09fUt21+/HPLbQclUMFeqIn+zzGQysNlssFqt6Orqwrp163jtrOF2KA0PD+e4fUWjUSiVSiQSCYjFYqxYsaKoNARJAXJbXAHQTrD81FYzc/rVQCD9OqLaSJ9b8JRKpTmCZ5OTk7y0W/JxDWLnZ7PZ0NbWBrPZTE8f3C4cbmF2OdEz0rIZjUZztG74ar0jkXU4HEZvb++SYaZ8cCdDSTGQOE1VmtpIp9Ow2Wyw2WzLtnsCQPof/gHShx+G5L/+C8lrrsn9JcPkkG3Q54P31CmqekleK+60an6+fmBggFdpCG6LKpGq4LZeqtVqZDIZhEIhmEymhg3PEXc0iURCxQ7lcjnVRIrH4/RkUkgaopBdJOmS4c4EtEpOv9VxTpC+QqGoKKdPBM/8fj/0ej2dROWCjyJsremdWCwGv98Ph8MBk8lEfQDIFyeTydAvDhE9m52dLVv0jGvKkUgk4HK5qK9rpUU3stHMz89DIpGgv78fWq22YsLL77rhTrqqVCpaCM6PmvONSsohPHbLFrBtbRCXSCEyDAPtY4/B9C//guR99yH0oQ/B7Xbj9ddfp0VXhmHgdruhVCrLllOuFSTnrtFoMDMzA6/XSx24IpEIfD5f2SeTapHNZmG32zE/Pw+NRpNzquCeTJZLAwFvSEPk20Vy5ZPj8Tjsdjt6e3trLrq3qr4QHzgnSL+clk2u4JlIJKJeuMUiB76i9EqvkV+YJRo0hARJCocQPV+iZ9ycayqVyvmikeN2oSEnbgqn0BBXLchPbQSDQbhcLirDbDAYIJVKYbPZihqVlEJ21SqIDh8GFhaAZWockoceAgBk3v1uWnTt7e3F7OwsrVUQ0iKdUvU83pP24ZmZGXqqIOkr7mAROZmQdApfHUrpdBoWiwV2ux3d3d3YsmXLkhRh/skkmUzC6/XmFKj1ev2S2lIhaQgA2LNnD1iWLWgXWelr3apFYT5wTpC+VCotmtMn+jJerxc6nQ6rVq0qi5QaPVhVrDBLpnq58gjEjrBeomfcaCuTydDR9HA4TLsuRCIR7WyqJWddLrjDMiMjI5idnaUpOLlcjt7e3oombwkyV14J8eHDkPz0p0h/+tOF/yidhujIEbBGI9DTsyRff/HFF1MhOp/PB7vdjomJiboZxLhcLszNzUEulxc8VeQPFuW3g5Kp4GpeL66EdiEdouVA0kCkQE26XE6dOkU3cZ1Ol3Oa5W4ADMNgcHAQo6OjSKVS8Hq9dJiPDEvp9fqy1nMmFIWrxTlB+vnKdtyoPpvNore3t2LBM74i/VKkHwwGYbFYEIlElhRmuYNp2WwW3d3d1Fw7EAigp6en7nlbbu91Op3G9PQ0Dh8+jEwmQzVLdDpdQ6KmRCIBi8UCl8uF7u5unH/++ZDJZEvkF4oVAgsh/dGPQvpv/wbxb35TlPTF//VfYDIZRC6/HIcPHSqar8+fUeAO8MhkMloHqKZoXktxlmvLSMiSO6lMaibLbUyRSASzs7MIhUIYGBioSUIbWDo7QSZ7X3/9ddrOq9fraUeaw+FALBajKSGJRAKTyUTtIok0xPT0dFl2kXzXBwp9z5u1qTAtnLvibWEsy2LTpk145JFHkE6nqaxDb29v1S2ANpsN2WwWZrO56nUFg0HYbDasWrUq5+fcAjI3L0s2LG63g0QiQTQaxczMDBU5MxqNGB4ebsgwEbC0v91sNkMul9PBK5/Pt2y+vVbkG5WYTKaiX1jSdul2uxGJRHKsAot9CRU9PUAqhZjHs/jv3l4AQOz0Z0By0UWQvfYaXn3iCZjOP7+qfD3JbXs8nopaVLmOXCaTCWazmbdNnhtt+/1+KBSKJa5aJIWUTqcxODgInU5XdzLjvocLCwvIZrNQq9UYHx/PIXHuKYD8GwA92bjd7qJ2kfF4HEeOHMHWrVt5WXMmk8G+fftwwQUXAABtU61iY6n5xT3rST+bzeLZZ5/FLbfcApVKhZ/+9KcIBAI477zzarquw+FAMpnEwMBA1dcgWjJk2Iw7Mdvd3Y3e3l56lGVZdklhlvR4E9Gz/v5+MAwDl8sFl2tR0YK0XNZjAyBkG4lE0NfX90Y7Yx6IiqPL5YLH44FCoahZ7phbGJZKpRgYGKh4XoKkW9xuNwKBQFHrw7Z3vhOS559H9MABYNWqRdJnWRx76SXYrVZcesUVYDUaxOfmqnou+SAtqm63m3a25OemuZF1f3//shsdH+AWXd1uN9LpNLLZLBQKRUNnOoDF981qtcJqtUKv16OzsxOBQAA+nw9yuZyeAkjRGii+ARC9I7KBEInp9vZ2TE1NYfPmzbysOZVK4dChQ9i+fTuA5pL+WZ3eOXr0KD7wgQ/gggsugFwux4svvgiGYbBv376ar81XTj8/1ZQ/McvtrSeF2VKiZ+SoTjpuSK6WbAC1TLgSQ5X5+fmiHreFnidRcRwbG8vpu5ZIJAXbG4uBKGwSeYZaCsOFJCFIDpmreS++/npInn8e0v/8TwS++lW0nS7EisViXDQ7CyaTQfqKK6paQyFwW1S5ImxHjx6lMydisRhDQ0O86BCVA4ZhoFQqIZfLwbIsOjs70d7ejmAwiImJCdo6W8ugWikQlU+n04menp6cegGRNCAuWNz6hF6vh0qlKtgNJBKJcvSOiMT09PQ0EokEpqameFGH5duFqxY0NdJnlnfOYgHgz3/+Mz7NyaVOTExg165duPrqq3HTTTfhhRdeoFOEDz74IDZt2kT/NpFIIJlMQqVSYdOmTfjLX/5CSZ/suNXC4/EgGAxiZGSkqscTaWKr1QqTyYS+vj4q0ZvJZGhUT8geAOx2OxU96+/vr2g4iUSPLpcLyWSS5jTLlQVOJBKwWq05KRw+unBisRhcLhfcbjdYlqVfwPxrczVhSBG5Xq2GZCDI5XLB6/VCnMngTZdfjpjZjAO/+AUufuc7wTAMYnY7ZJddBtGePYi9+iowNla39bhcLszOzkIsFkMmkyEcDhdMt9QD+ZH1wMBAzgZNJBhItKxSqWi0zcd7RIrDXq+XpmXLSRGS+oTH40EoFKInufyZjkKngFAohMnJSej1eqqbQ8zUi4kDLodoNIrjx4/Tk8M5m95hGOYbAHwc56wulmWJicqShfl8PoyNjcFisUCpVOKmm27ClVdeiZ07d5a816ZNm/DCCy9AJBJh7969Nad3vF4vFhYWMDo6WtHjuIVZvV6PQCCATZs20W4E8n6QidlYLAaLxUJFz/r6+mr+gpOcqNPppC2XRqOxYGEzEAhgfn4ekUiE5svr1YVDfG9dLhf1KpXJZDRV1d/fT3veGwGihzM7O4uL3vc+yHw+/OW55/Cmq64CIxIhbrVCodUCHR2IWSy835+Qrc1mg1arRX9/f8Eed65hTSVmLKXAVdrs6emB2WwuSeLcaNnr9dZkXBOLxTAzM4NgMIiBgQEYjcaqo2WifOnxeOi8AtmYyOkFeGMDIF7Q69ato48nqUCfzwelUlmRXWQ4HMbk5CQ2btxIryeXy6v5LJ/x6Z2KnLMef/xxXHHFFVV9qNva2pBKpegbVGufdDWTvVarFXK5nBZmU6kUAoEAnZglUT3DMHUVPeNqt5AJ1+npaUSjUXocjsfjsFgstD+/Hvrn+SAzBz09PbBYLJg7nSMnR/B6RrNccFNIer0emzZtguSyy8Ds3o2tXi9wOi136tvfxoZUCrG3vpXXvntucbZYB1ah6VuuUXwtUhWxWAyzs7NYWFhAf38/zj///LI/e9xU3ujoKGKxGDweT45xTal1hcNhTE9PIx6PY2hoqGJXsELgKl+Oj48jGo3C4/Hg6NGjOWboKpUKCwsLmJqagk6nyzlxc7WFuHIXpJtoOUOeVpoWbnakv8CyrOb0fzMA/OTfKBDpX3rppfjMZz6DK6+8EgBw00034aWXXoJMJsNll12Ge+65p+iu+5a3vAW/+MUvoFarsX//fmzZsqWmNyEQCMDhcGDlypVF/6ZUYTaVSuHgwYPQ6XQwmUxQKpWw2+2w2+3o6upCf39/w0y9gcUj6KlTp+D1emnk2Nvb2xDCB3KNSrinGpLXdrlcVHqBKDDyua78/noiHQAAor/8BfIrrkD66qsh/vOfAQCZdesgfvFFvPZf/wW30Viz2iUp7NdanCUbudvtRigUom2XpdISoVAIMzMziMfjGBwchMFg4PX1zV9XfuF8YWGBzp0MDQ01TJQsnU7TulogEIBUKkV/fz/MZnPJNBDwxsmZfD6JNAS3U21hYQFWqxVr164F0NxIv+6kzyzvnPUQh+TBMIyfZVmiYJazMLvdjg0bNlAdHPIzk8mEZDKJj33sYxgdHcWdd95ZcB1vf/vb8cMf/hBGoxGvvPIKNmzYUFO+kRz/Vq9enfNzYnrOLczq9Xp6MiDtlgDolKbVasX8/DwSiQTUajWGh4erkiioFoFAAHNzc4jFYnS6l5w2nE4n9T01Go28F+pIDzXXqIQMdxUCGb13uVxYWFhAZ2cn/YJVa1K/sLCAubk5pFIpDAwMFCU7hVoNVqsFc3q6G/E4IJMhZrfnSAL4/X4qwKbX65dNhxE7zdnZWbAsi8HBQV7fe27bpc/nW2LKTu4/MzMDhmFqUo2tdF2kd550A8nlcoyNjdXkGV0pyPd1enqaGr5kMhn6epFp5WJpoPwNgPu8SBqJTIYvLCxQvjir0zt8OGcBwKOPPor3vOc9Ocfcnp4eAItpgZtvvhnf+ta3iq6DDOkA9ZmmTSaTsNvtcDqd0Gg0GBsbo1F6JpNZYiieL3q2Zs0aqNVq+Hw+OBwOnDhxAlqtFkajsaox8lIg+WqSwhkYGFhyH3KcJYRGFCVLuV1Ve/9y2v7ypRcCgQCVhe7o6KAbQKkNnXQhzc3NQaFQYHh4uGR/fXZsDKKJCUClAjIZMIkE7dopJAlBukAKDV5x769UKjE2NsbbxDQX+UNOpHPqlVdeoQN+nZ2dFXv91gqGYZBMJhEIBNDV1QWj0YhwOIyZmRlMT0/nqIPWYwMistIzMzPo6OjA2rVrc9LG5LNIevq56SnSJsrtBuJuAlwDFTJURtJVpMbRCO2lYmh2Tr+kcxbBI488grvvvjvnZ3a7nZoUP/HEE7ToUgh8WyaSa5SamCUbg0QigVgspm1gxUTPSHGIFI6IrSD5YtQahXGnVg0GQ0nRNfJcuYTGdbuqdOgqf5CrnPsXA8Mw0Gg00Gg0lNCcTidmZmZyiJYbKOTn6yvxm83u2AHxxATYVAo4XYdJ/cu/FFwXV+aAm/8FFmsq0Wi04vvXCtJ22dbWBpZloVar6RDdkSNHKNFWI79QLrgCbF1dXTnPX6/XY2hoKEdJk9s1w8cpkwQbc3NzUKvVJV9/7rRyOp3O+U52dnZCr9fT9NRydpEKhQJ+vx9qtRoWi4UOkzUDzc7p6wA8CmAAp52zWJb1MYvOWft+9KMfAQBmZmZw8cUXU59UgksvvZS2+m3atAn3339/0WjlAx/4AP7pn/4JGzZswOuvv46RkZGquxwymQwsFgtmZ2eh1WqXTMym02n6xpMUTr7o2XIpjHyQDYCkWirdAEhEPD8/n5PCqbULhyty5vV6oVQqYTQaC6Y0uKbmpFhbTy2eSCRCUwdisRgajQbxeByhUKj6ls/paSjWrQPEYiCTWezacTrLeihpO3S73Whvb0cqlaIKnOVKQtQCrlmL0WhEf3//ks2Q5NvD4TAvblxccNs+DQYDBgYGyirM56fNOjo66Am0ksE+7maj1WoxODhYVtdNMXBPc16vF1KplK5LoVDkZAFEIhGcTidisRhGR0fp76rc7Fs/p18DeF3YRz7yEdxwww3Ytm0bjh49ioGBgYqPs1zJZa1Wi2AwiC1bttDCLHfwQyKRIJPJ5Iie9ff313yE56ZaAoEAzbUXKnoRTRKLxQK5XF4whcMXuJE2mboluXFiwD4wMNDQWgXZ7KamphCLxWhqrZYhNUV3NxCJgAGQvuIKJB9/fNm/D4fDVMcmv+2QEK3L5UIkEqFidXzm1OPxOGZnZ+H3+8vucS9EtJWIlXGRP1BVTttnMSyZnzidKiHG7IVANIksFktFm02lIPpObrcbyWQypxuInCxGRkZgMBgALAYl3d3d1dxKIP1y8c///M945zvfiTe/+c2YmJhAb29vWXm1YoXZTCaD1157DevXr6d/S3rriXY7ET3r7e2ti+hZsQ1AoVDAarXC7XbDYDBQLZxGgQiv2Ww2sCyLjo4O9PT0VC0mVim4U8PczQ5YOqRWaaQtu/xyiF56CQyA+LPPInvRRUv+ppribDabpZF2IBCgBep8SYhyQfLj0WgUg4OD6O7urmoj4fbdezyesmWYiXeDz+er28kuHo9TDZ5kMknTQGq1GtlsFhaLBTabjXdNolIgyrOEAxQKBWKxGDZs2AC73Y577rkHnZ2dePjhh6u5vED65eJf/uVf8KY3vQmXX345Tpw4QSOqYsgvzPb19eUUZtPpNPbv34++vj4YjUbI5XJ4PB6qnd7f398Q8SkCcnwmXTBqtZoKYDWqPzjfqIRsdtFoFE6nk6ZajEZj2UMtlSA/X9/f378sMZFWPZfLRWcUSumvi3/4Q8j+z/8BCyAWieT8Lt/cfHBwsKqTHVcV0ufzVaRVRDpx6tEJBOSKlZH0lMFgoJtmNBrF7OwsLwNVlYC09TqdTvh8Pqo6Oz4+3rD5DuCN0/Xc3By0Wi2VQ/na176Gp59+GsFgEO9///vxyU9+EmPVTXALpF8ubr/9dqxbtw5XXnklTp06RTsacm54Ok9HPGN7e3thNBppYTZ/YjabzVI5W0K0Y2NjDRefIl0wpOjU2dlJTwALCwtQq9U0BcT3F5BEtXNzc0in0+jv74fBYCh6HyK74HK5wDAML4JwZLPx+/1V5+uJ+JrL5aJ1k4Km5+Ew2t77XmS3b0f6y18GsLh5kMnZcjabSkAGgYhYHZEF5kbahQzOG9EdwlW7DAaDtHg5OjrKe49/KSSTSVozMZvNaG9vh9frzWmbrEegQcAle51Oh8HBQUilUhw6dAj33HMPkskk7rjjDgwNDeHpp5/Gb3/7W/z85z+v5n0SSL9cfPWrX0V/fz+uvfZaTE1N0co78AZx2mw2yOVymM1mGu0VK8xyRc/6+vqg1WpppJHNZmE0GuumbgmATsy63W50d3fDbDYX/ECTbhun0wm/34/Ozk4Yjcaa2i2BN16z+fl5tLe3082mEhBBOJfLhWw2C4PBQNNTpUCiYeK5ulx/faUoNguQn2rhDpP19vair6+vrtaDwNJIWyaT0XoA6RJpJBYWFjA9PQ2WZaHVahGNRhEIBCo2iq8W3DTSwMBAwYE20jbp8XgKnk5qAbdATHSJpFIpDh48iLvvvhvZbBZ33HEHLrroIr42QYH0y8U3v/lNqFQqfOADH8DMzAyUSiU6OjpoYZZMn3InZrmFWfJlL0f0jJCZ83RnR3d3N4xGY81RBiFwMshFunDKJW8+NoB8o5Jim02lSCaTdANIp9NFPXiXy9fXA/mpFqVSic7OToRCIcRiMfT39zcshUFArAhtNhuUSiV1TtPpdOju7q5KeqESkDrXzMwM2tralpwsuJ0tHo+nLpE2kYoIBAIYHBykA4WlkO+nUG2XEjnlz8/Pw2AwYHBwEBKJBAcOHMA999wDALjjjjtw4YUX8v1eCKRfLr773e+CZVncfPPNmJiYQDAYpBK2y03M1ip6lkgk4HQ6qWgYOQFU8uHPZDJwOBywWq1QKpXo7++vmei4g03E5MRoNBatAVRiVFIriAevy+WiOjJarRaBQKDsfD3fIN7EU1NTSCQSYBgGCoWC1icakTfOtyLkprHypSqIFACfKT2u2md7ezuGhobKkgkhkTZpr65l8CoajVIP3aGhoZpOd/ldSvnTysUeY7VaYbFY0N3djYGBAUgkEuzfvx933303JBIJvvSlL+G8886r18YrkH65uO+++/Diiy9Cp9PhhhtuQFdXF4aHhwEgh+y5UX2+6BnZHKpFPB6nJwCRSEQ3gGKEwU0f8BlV56PYBtDV1QWv11uTUUmtCIVCOHnyJAKBAC0C9/T01HWAiAtucba9vR2Dg4O01TcajdJZAJFIRE8nfG9G+VaEpU4WRHrB5XLRlstyJ5WLXY87UDU4OFj1cySDV263G7FYLGfwarn3k4iwJRIJDA0N8d4kwZ1W5mpP6fV6tLe358wZkDkHiUSCvXv34p577kFbWxvuvPNObNu2rd6fS4H0S2Hv3r343ve+h7/97W/Yvn07vvnNbyIajSKTyVBjb1KAkkgkSKfTsNlsdRc94xY0uRsA0eiYm5tDMpmsOIVTK7gRbTAYhEKhQH9/f9ka5nytoVC+nrQ1Op1OasJeL5mKSouzpE/b5XIhk8kUTU9VAj6sCLmuZV6vF21tbWV78ZIhRJvNhu7ubvT39/N6osl3LitkFB8MBjE9PY10Oo3h4eGGibBxZb7D4TCy2Sx0Oh3Gx8chk8mwZ88e3HvvvVAoFLjzzjuxZcuWRgVDZz7pMwxzHYC7AKwGcB7LsvtP/ypnYc888ww++clPIpPJ4KMf/Shuu+02AMD09DSuv/56eL1ebN26FT/72c9yPpgPPvggxsfHYbVacfjwYXz2s5+F3W6Hz+fD8PAwlEolxGIxQqEQ1Y3v7e3NUVisN2KxGB3iSqVSaG9vx+joKHQ6XUPuT5BvVNLT00PbLb1eLzo6OmgKqB4bQCX5erIB5Hfb1EoK8Xh8iWFHpZ8DEs06nU6aay/mV5APki+fnZ2FRCLB0NAQrzULruUhiWbzTWtSqRTm5uYq0tGvFVyjeDJ4lU6nIZPJMDo6Wte6TSFwNzyj0QilUolnn30Wd999N9ra2tDV1YX77rsPF198cUNPvjhLSH81gCyA/wDwfwqRfiaTwYoVK/Dss8/CbDZj+/bteOSRR7BmzRq8973vxTXXXIPrr78e//iP/4iNGzfi1ltvXXKfhx9+GP/93/+NH/zgBxCJRDk5Y5Zl6dG9UZEEAakXeDweWlQleUapVEpzxvUaLCGR/dzcXE4aK/81IBEjmbhtb28vKrlQKVKpFGw2G2w2G3Q6HQYGBipKH+QPqVXTohoKhTA7O4toNIqBgYGKZDKWA5ETdjqd1Ii9u7t7yemEqwmjUqkwODhYd1ltbjSbSqWox0M4HIbZbEZvb29DC9Sk/Xd6ehoMw1A7RuANTap6vyaZTAbz8/NU18tsNkMsFuOvf/0r7r33Xmg0Grz73e/GiRMn8Ic//AH/8A//gI997GN1XVMeznzSJ2AY5nkUIf2XXnoJd911F37/+98DABVeu+2222AwGOBwOCCRSJb8HRd//etfcffdd2NmZgabN29GMpkEAHzhC1+ARCKBz+eDSCSCyWSiaZZ6gXy4iRaP2WwuSDKkP9vlckEmk9ENgI+oi0hEWCwWdHZ2ViRLwd0ASmnuLIdYLIa5ubma+usLrY3bobRcgZpseLOzs2AYpu6bfv4sgEajgU6nQywWo2mkfCvCRoAUR/1+P9ra2nJMRRoRBLEsC4/Hg5mZGSgUCgwNDeV8FhOJBB2iSyQSZQ3RVQrSEWW329Hb20u19P/nf/4H9957L3Q6Hb70pS9hw4YNOfdMp9MNywicxrlB+o8//jieeeYZEAG2n/3sZ3j55Zdx11134YILLsCpU6cAAPPz87jiiivw+uuvF7zHX//6V3zjG9/A9PQ0BgYGMD8/j8suuwzXXXcdNmzYkNNpI5VKYTKZeCNZ4A2itVqt6OjooC2f5SASidCpVplMBpPJVJUWSjGjkmqRr7mjVCppzrjYBrCwsFCX/vpCa+Nqm5OCplarhcfjwdzcHDo6OnKKs41CPB7HqVOnqNVhV1cXTCZT3VJnhRAKhTA9PY1kMonBwUF6wiMCfy6XixdJiGLgdgN1dHRgaGiopB5SfpdSvhFLpUin05ifn4fD4aBm9CKRCH/5y19w7733wmAw4M4778S6desancYphtbX0weWN1JhWbaonDLfCIVC+OpXv0p9KmOxGJ566il8+9vfxqlTp3DFFVdg586d2L59O6LRKBwOB/bv3w+FQkFJtpoPViwWw/z8PLxeL0wmEzZv3lwx0ba3t2NkZAQjIyM5EsLlrC3fqMRsNmN0dJSXozvDMFCpVFCpVBgdHV2yNnICICm1ubk5OjVa7zxtvvQySR28/vrrkMvltEDdyEgt34pwzZo1dNjP5XJhamqqItmFakCkGgBQhyouRCJRjjUg2TgnJyd5WVul8sZciMViOsXNNSw5deoU9a0txyieKwTX19dHWyxfeOEFfOMb30BPTw++//3vY+3ata1C9rzhjIj0+UjvlEIwGMRvfvMb7N69Gw6HA+9617tw7bXXYmhoKCfK7ujooBHZcqRZqTxBNSgUZXPTLNUalfC1tkgkQu0fM5kMNBoNxsfHmxJVc4uzPT099FTn8XjQ1tZW99pJuVaEhWQXCMnVeiLjDlQNDw9XrAuUvzZCwKXE1wi4rZ+kdsNXKovrW5vfQlusSE3qFgzD4M9//jO++c1vwmw244477sDq1atblezPjfROOp3GihUr8Mc//hF9fX3Yvn07fvGLX2Dt2rW47rrrcO2119JC7oYNG/BP//RPNa3F6/Xil7/8JXbv3o1oNIqrr74a1157LUwmE0KhEBwOB3w+Hzo7O2EymXIKhvkpnIGBgbo4IuWDbAAOh4N2ZqTTaRiNxooLo3wgP1+vVqvh8XjohCaf9Yli4BZnCdEW2nS52vt8kSzwxsZfixVhIa0ig8FQttwCy7I0qi43hVLJ2riSEKTYmr+pc+WNyUBTvRUviQInt0idTqcRDAbpCQ8A/vjHP+Kb3/wmhoaGcPvtty+xP21BnPmkzzDMewB8D4ABwAKAQyzL7rDZbOxHP/pRPP300wCAp59+Gp/61KeQyWTw4Q9/GLfffjsAYGpqCtdffz18Ph82b96Mn//857xGD06nE48++igee+wxiEQiXHPNNbj66quh1+tzCoYdHR002uAjV14NuEYlRP65nGlbPkFmDBKJBO2CySc5oolO6hOFHK6qRX5xtlKizZ+fqGbgipurViqVSwqT1YLbbbOcVAXwhkyAxWKhao/13Pjzh650Oh20Wi1CoRD1siYDTY1EMpnEzMwMnE4n2traMDs7i6eeegpr167FH/7wB4yNjeH222/HqlWrGrougsceewx33XUXjh07hr1792Lbtm0F/460rJ84cWISwI9Ylr0HABiGGQawC4AOwAEAH2RZNrncPZtO+sugpRbGsixmZ2fx6KOP4le/+hU6OztxzTXXoK2tDfv378e1114LqVRK+7JNJlPdNVDIukhRspBRCdFBIZ029dgAiN8vydcPDg6Wna/nps5qSbMQlcP5+XneWh7JBLXb7aZSvfnpAi5IRGu1WmueXC0FrlRFPB6nvgAKhYK2v9ZjoKocxONxnDx5Eh6Ph9YH8k/E9UYymcTs7Cy8Xi/6+/upn/bTTz+NH/7wh1hYWEAsFsPGjRvx4Q9/GDt27GjIuvJx7NgxiEQi3HLLLfjWt75VkPS5Leujo6MyAPsA/D3LskcZhnkUwK9Ylt3FMMz9AF5lWfaHy91TIP0qEAqFcN999+GBBx6ARqPB6Ogodu7cif/1v/4XFAoFvF4vHA4HwuEw/cDzbYdHJodtNhs0Gk1Zk8P5cgu1Km5yp1Z1Oh36+/trUnnkplnKnVHgaugbDAb09/fXpeVxOUG4UlaE9QbRZpqdnUUsFkNnZyeGh4cb6ucA5Mobc1MofEpClLOGmZmZHNVNYDFSvu+++7By5UrcfvvtGB8fB8uyOHz4MBYWFvDWt76V97VUgksuuaQo6efVKhmGYb5w+lf3AHADMLEsm2YY5kIAd7Esu+wO1mxj9DMSu3btglKpxGuvvQatVovDhw9j165d+Pa3v42VK1di586duPzyyyGVSuHxeKhdn8FggMlkqikCJblyn8+Hnp4ebN26tWyCye9mCQQCcDgcOHnyZEUDTdw1kBoLH1/g9vZ2DA8PY3h4mE4CHzx4EBKJJEemgrsGYgN43nnn1bXVsa2tDWazGWazmUbZx44dQ+S0kQqfr0Ml4NohEpLjGter1WrablmvKDtf3vj888/PuRfxruBKQpCCMl/qm4lEAjMzM/D7/RgcHKQGJU8//TTuu+8+rF27Fj/72c9yjEsYhqGdfK0Mq9WK/v5+7o8sAM7HYkpngWXZNOfnfaWuJ0T6PCKbzWLfvn145JFH8Nxzz2HLli3YuXMn3vrWt4JhGLjdbjgcDqRSKboBlKsdX89OoPyBJrVaTY/j3EixnHx9PUCEzYhSKfnMDg0NNWwNXHCtCM1mMwDQXDZJs9RbEI6IsHG9dwtNUZMNwOfz0SibjylqYHHjnZmZQTAYrEjemIArWAeAnp4qKTTnk73JZALLsrQVe8OGDfjiF7+IkZGRip8fX7j88svhcDiW/PzrX/863v3udwNYPtLPm1NiGIb5IBZJ/y4Ae1iWHTv9i34Av2NZdt1y6xEifR4hEolw/vnn4/zzz0cmk8H//M//YNeuXfjiF7+Iiy66CDt37sSFF15INWaOHTuGTCZT1HAl36hkZGSkLo5IZDioq6uLbjBOpxPHjx+HWq2GTCaDz+erOF/PFxQKBTo6OuD1egEASqUSoVAIFosFqVSKl06bcrCcFSER7/N4PJidnc0xOudzcpQ7UDU0NLRsa2H++0qi7Onpacjl8qoL6JFIBDMzM1TeeNWqVVU9P1LkHhoaokXq48ePU+9iYixe6NrxeBwzMzNUT3/FihXIZrP4zW9+g+985zvYsmULdu/eTZV0m4nnnnuupsf39fVhfn6e+yMzACsALwANwzCS09E++fmyECL9BiCVSuGPf/wjHnnkERw4cACXXnopdu7ciS1btiCVSuUYrphMJmg0GjoZXE9J5eVAxtKJ5y9RGTQajQ2TVy5VnM3vtCFmNXxuAFwrQrlcjsHBwbI2XtI55XQ6EQqF0NXVVdNrxx2oGh4ernnegsgIkzZVEmUv9znjyhsPDw/z7r9LQLyL3W43VVMl8svJZBLT09MIBoP0pJfNZvHkk0/i//7f/4utW7fiC1/4AgYHB3lfVz2xXKTPbVkfGRkhhdz3syx7hGGYxwD8klPIPcyy7A+Wu5dA+g1GPB7HM888g0ceeQQTExN4xzvegeuuuw5r167F888/T9UFicqk0WhsaFGQmyvv6emhFoBE1MzhcCAYDNZV1phbGC23A6XQBlCpWQ0X3EEijUZTkxVhviBcuQYnXE0auVyOoaGhusx8kH57l8sFlmWXdCkReeNMJkPljRsFIglhs9mozn1XVxdGRkagVCrxxBNP4Lvf/S62bduGL3zhCxgYGGjY2vjAr3/9a3ziE5+A2+2GRqPBpk2b8Pvf/x42mw2FWtZPnjw5BeAnLMt+HQAYhhnBYsumFsBBADewLJtY7p4C6TcR4XAYv/71r/Hd734X8/PzMJvN+MpXvoK3vOUtiMfjNNonWjvL6dnUCqJfT/L1y9UMyBfR6XTSDYCPFtX84my1Gv5csxqGYSpyK+MKb9Wj5THf4ITo2nBbaEn6b3Z2FiqViteBqlLgzgLE43GwLAuZTIaxsbGGTXNzEYvFMD09jXA4jKGhIUilUjz55JP4xje+gVQqhVWrVuE73/kO1q9f3/C1Efh8Przvfe/DzMwMhoaG8Oijjy7ZGP/85z/j05/+NP33xMQEdu3ahauvvho33XQTXnjhBZo2ffDBB7Fp06Zitzvzh7OWQcsujC84HA7s2LEDl19+OW644QYcPHgQu3fvht/vx9VXX42dO3eir68vp5ddqVTyJsxFhohIf301Mg1cY5NQKERnFCopZAaDQczMzNSlQEw2AFIEJlFsfv1kOSvCeiG/hba9vZ2a6FQjMc3Xmnw+H2ZmZiAWi3P8gBvlwQvk2iIODw/TYcNf/vKX+P73v48LL7wQ11xzDfbs2YOnnnoKt9xyC2688ca6rqkYPve5z0Gr1eK2227DPffcA7/fj3vvvbfo3/t8PoyNjcFisUCpVOKmm27ClVdeiZ07d5ZzO4H0z2SQCd78aU2Xy4XHH38cjz76KDKZDN7znvfgPe95DwwGQ47Wjkqlgslkqrgdj9tfT6Y1a+mvJyAKiMTZSq/Xw2g0FtwAuENlEokEg4ODdY8kibuV0+kEy7LU2MThcJRtRVgPcOsnMpkM6XQa7e3ttNOmEW2g+fLGw8PDOfWTfHVLYlqj0Wh4fb0ikQimp6cRi8XorEEmk8Fjjz2Gf//3f8eb3/xmfP7zn6czAATZbLbh7xvBypUr8fzzz6Onpwd2ux2XXHIJjh8/XvTvH3jgAbzwwgv4r//6LwAQSJ+DogurdHS5UretVgHLsrBardi9ezd++ctfQi6XY+fOnbjqqqvQ1dWVEyUWa7Pkgqv22dvbS/P19QAhCYfDgWg0SjcApVIJh8ORo+Nfb2OMQnC73ZicnEQsFoNMJkNvby9MJlNDI+tkMon5+Xm4XC6q4S4Wi6mOEhE241uqgot8eePh4eGSAQCpUbhcLiwsLBRMUVUKLtmPjIxAq9UinU7j0UcfxQ9+8AO89a1vxec//3k6WdtK0Gg0WFhYALD4enZ1ddF/F8Kll16Kz3zmM7jyyisBLJL+Sy+9BJlMhssuuwz33HPPcqnIc5P0Kx1drtZtq5XAsiwmJyexe/duPPHEE+ju7sa1116Ld77znejo6KBF1kAgQLXZSZG1knx9PUAmRufm5hCNRqFWqzE6OtrQgiBQ3IowkUjQFFA5cgu1gjtQRSQClns/8gXhyKRyLYEKUWCdnZ2tSTIi37OAOKqVO3HL7QgaGRlBV1cX0uk0du3ahR/+8Ie49NJL8bnPfY5O1jYLy/Xa33jjjTkk39XVBb/fX/A6drsdGzZsgM1moxs40SZKJpP42Mc+htHRUdx5553FlnJukj5BBaPLdZFjbhZYlsWRI0ewa9cuPPXUUxgdHcW1116LHTt20J56h8MBv99PrSDJF6rR4OrHm81mGAwG2snC15RyKVRiRUjkFpxOJzKZDG0D5WMD4Pa3Dw4OVlW74A40VdOlxBViq4e8MUk/EhN2IvWdv0GFw2FMTU0hlUrR9s9UKoVHHnkE999/Py6//HJ89rOfhdFo5GVt9UQl6Z3/9//+H44cOYIHHnig4O+ff/55fOtb38JTTz1V7HZnholKM5A/umw2m/Hyyy/D6/VCo9HQKMRsNsNqLTnP0FJgGAbr1q3D1772NXzlK1/BwYMH8cgjj+Duu+/GqlWr0NnZiaNHj+L73/8+VCoVfD4fjh8/nqMDVG+Q00UymcTAwABWrlxJCc5kMsFkMiGdTsPtduPkyZNIJBIwGAwwGo28bQCZTAZWqxVWqxV6vR6bNm0qSXBcuQWyARw7dgzpdJoSbKWdNKRQnUqlMDg4WJMmDnegiRSpX3vttYKtllxwX4vu7u6K5DvKBddQZ2xsjJ5QDh06RLX3lUolLBYL0uk0DUSSySQeeugh3H///dixYwf+8Ic/oLu7m9e11RNXXXUVHnroIdx222146KGH6JRtIZDvKRfEj5dlWTzxxBNYt27Zgdqa0bKkX87osoDFKeCtW7fCaDSCZVk8+eSTGBwcRDQaxQ9+8APs3LkTF198MZWBOHXqFBKJBI1g+WwFJMXA2dlZSKXSksVZiUSCnp4e9PT0UD2bEydOIJlM1rQ+bq68p6enak2c/A3A7XZjYmKCCq4ttz6ulr5IJKISz3yCzHIMDAzQFNXRo0fpCYVMKpMWVJPJ1FB9IK6WksfjoZu7TCbDnj17IJFI8MQTT+CBBx7AFVdcgeeeew4Gg6EhayuEclovgUX3LtIiOjAwgJ/+9Kd473vfi/vvvx9+vx86nQ6zs7P41Kc+hR//+MfU5nVmZgbz8/NLxN0+8IEPwO12g2VZbNq0Cffff39dn6eQ3jkD0zuF8PLLL2Nubg7vec97IJFIkE6n8fzzz2PXrl3Ys2cP3vKWt+C6667D9u3bkclkqA4QiWBrKWJyTdY1Gg0GBgZq2kzIBsBdXzkplnwrwlK58lrWR2oAZIMiipuNGqhaDslkkk4yx+NxdHV1YXR0tOHyGcDiKWdqagosy9JJ4mAwiC984Qv4n//5H0SjUdxwww244YYbsH79+qa6VZXbetnR0YFwOLzk5w2qFQo5/XJGl4u5bb3jHe/Ahg0bkEwmsXHjxnoPVTQNyWQSv//977Fr1y4cPnwY73jHO7Bz506sX78e6XSa5rCz2SxMJlPZOeJkMgmLxQKn0wmj0Qiz2cx7FxRXpqLYBlCuFWE9wN2gotEostksNBoNxsbGGjZQxQVXR57UUIjJORGEI62q9XyNAoEApqamAAAjIyO0YP7www/jxz/+Md71rnfhM5/5DKRSKZ5++mk899xzeOCBB5rWdgmUn5svRPosyzYqmDw3Sb/S0eViblsnT56E2WzG/v378Z3vfKfeQxUtgWg0iqeeegq7d+/G1NQUNYNfuXIlkskknE4nnE4nRCIR3QDyc7/RaBRzc3O0ONvT01NXWWOC/CKrSqVCJBKBWCyuyoqQD3CNUzQaDRQKBXw+H5LJJNWzaUQNpZDEcj6BptNpOkcRjUbrMmxFyJ5hGCoQGI/H8dBDD+EnP/kJrr76anz605+GVqvl5X58otzWS4lEgk2bNkEikeC2227D1VdfDY/HgwsuuACnTp0CAMzPz+OKK67A66+/zvcyz03S5wsNHqpoOQQCAWoG73K5cNVVV+Haa6/F4OBgjgyEVCqFyWSCTCaDxWKhxdlGRtQEpK+ciI8Bi3WNYkql9QJXrqGQcUoqlYLH44HT6UQ8Hqc1AL43gGrljfOHrYgiaLUb58LCAqampiAWizE8PEzJ/sEHH8RPf/pTXHPNNfjUpz7VlA4yLvhovbRarejr68PU1BQuvfRS/PGPf4RarRZInwfUfWENHqpoaXg8Hvzyl7/Eo48+ilgsRs3gDQYDfve731FjaZVKhf7+ft402cvFclaEpIhJlEor0dqpFFx3qL6+PvT19ZV8HUiXEkmxcLuUqt00SftnNBrF0NAQ9Hp91dfK11LSaDRUEbRUusXv92NqagoSiQQjIyNQqVSIxWL46U9/ioceegg7d+7EJz/5yabo9lSKSoNA4I3Aj3xXhPRObeBlYS00VHFGgGVZOBwO/PznP8cDDzyAcDiM9evX43vf+x7MZjPtw3a73ejo6KA6QPXKxVZqRcgVW+NTbpnoty8sLBRNn5QDIhtM5hQqzbFz9fTrIW9MBOGcTiedti1kqenz+TA9PQ2pVIqRkRF0dHQgGo3ipz/9KR5++GFcd911+N//+3+fEWRP8NnPfhY6nY4Wcn0+H77xjW/k/I3f74dSqYRMJoPH48GFF16I3/zmN1izZg2tFZJC7oYNG/BP//RPfC9TIP1a0OChijMGL774Im699Va8973vxRVXXIFnn30Wv/71r6HRaHDttdfiXe96Fzo7O3MM1zs7O3k1v+bmqKtV3OSmqLgpoEo2AD4Gqoqhkg0gEAhgenoa2Wy2YfLGhZy3lEolfD4f5HI5hoeHKdn/+Mc/xs9+9jNcf/31+MQnPtGUTqFa4fV68d73vhdzc3MYHBzEo48+Cq1Wi/379+P+++/Hj370I/ztb3/DLbfcApFIhGw2i0996lP4yEc+AuCNWqHP58PmzZvx85//vB6nTYH0a0E5OzvBBRdcgLvvvhtve9vb6M+4QxW33nornnnmGUil0op6fJ988kkAraUHFI1GIRKJcvLjLMvi+PHj2LVrF5588kn09/fjuuuuw9/93d9BoVDk2C1qNBpqBlMpQXKtCPkk2VgsRjeAQp67+SAa8ul0GkNDQ3UzDCEgrlvcIqvRaEQmk8H09DREIhGGh4ebQqZEvuLUqVPIZrMAgCeffBJDQ0MIBoN4/PHH8f73vx8f//jH6+LsVi7K6bM/dOgQbr31VgSDQYjFYtx+++143/veBwBnSjeeQPq1oJydHVgcqrj44osxPz+fE8VeeumldKgim83ife97H/71X/+1lXt8eQHLsjh8+DB+8Ytf4He/+x1Wr16dYwafb7ZSjtb+claEfIO7AUilUqplI5FI6j5QVQ7S6TRmZmZgsVjopG1/f3/dfXfzQch+eno6R3kzFArhG9/4Bv7yl7/AZrNhy5YteN/73odrrrmmKW2qBOX02Z84cQIMw2B8fBw2mw1bt27FsWPHoNFozpTGDIH0WwVnSI8v78hms9i7dy8eeeQR/PGPf8TWrVtx3XXX4S1veQtEIlGO1j5XBoJhmKqtCPlENBqFw+GAzWZDKpVCe3s7VqxY0RSy5w52EbkFuVyeI1dNTgD11LTPX8fw8DD1JX7ggQewe/dufOhDH8I///M/o729HUePHsUvf/lLfPzjH29qd041hdiNGzfi8ccfx/j4uED6LYCWXVghnCE9vnUFMYN/5JFH8Ne//hVvetObsHPnTlxwwQUAFjuEyBCTXC5HLBaDVqutyYqwFnCF2Ig8cCAQgNvthkwmyzkB1BNceWPilFVMP4fbZsn3BkA24ZmZmRyZ5VAohP/4j//AY489hhtvvBG33nprU+SwS6HSbry9e/fixhtvxJEjRyASic6UbjxBcK2RWK4TiAuGYYp+CWdnZ3N6fNevX39GFr0KQSwW45JLLsEll1yCVCqF5557Dj/72c/w6U9/Gpdddhn+7u/+Dn/84x8xMDCAbdu2QSaTIRgMwuFwwGQyNYz4SfunxWJZIsSm0+kwMjJCu5T2798PhUJBNwA+21SJ8fvc3By6urqwYcOGZecMiGgZMQP3er2Yn5+nffbVehZzyV6lUmH9+vVQKBQIBoP43ve+h8cffxw333wzXn755aambwB+voPAYj3ugx/8IB566CGasr377rtzuvHuvffeM74brxAE0q8Azz33XNHfGY1GWtgl/qqF0NfXB2BxNP2SSy7BwYMHce2112JhYQHpdBoSiQQWi4X+3ZkKqVSKK664AldccQVmZmbwmc98BjfccANWrVoFlmVx0UUXYfXq1VQG4tixY8hkMnUdskqn05ifn6ebzLZt24oWcjs6OtDR0YHR0VGEw2E4HA6a7iBywdVuAPnyxlu2bKm4aC8SiWAwGGAwGOgGYLFYcOzYsbI3AO6gW2dnJ910AoEAvvvd7+JXv/oVPvzhD2Pfvn1NOYkVAh/fwWAwiHe+8534+te/Tk+hAKhBi0wmw80334xvfetb/C6+RdA8oYuzDEReFUBReVW/349EYtGo3uPx4MUXX8SaNWvAMAze9ra34fHHH1/28WcistksPvnJT+Laa6+Fw+HAM888gy1btlAD+G9/+9uIxWLYvHkzNmzYAIZh8Prrr2P//v2Yn59HMpmseQ3JZBKnTp3Cvn37IBaLcd5552F4eLhsaeGOjg6MjY3h/PPPx/DwMEKhEPbt24fDhw/D5XIhk8mUdZ1MJoO5uTm8/PLLSCQS2Lp1K8bHx2vu0iIbwLp163D++edDp9PBarViz549mJiYoL4KBGQWY+/evfD7/di4cSNWr16NeDyOf/u3f8M73vEOaLVa7N27F5/+9KdbhvBLoZzvYDKZxHve8x586EMfWpK7t9vtANAwieNmQcjp8wS+enzdbjei0ShUKhWGh4fP1rYzAIub4K9+9Svs3r0bwWCQTgH39vYikUhQHSCJRAKTyVSxXSBXdbOWgapCYFkWoVCI+hV3dHTQE0AhzRsi2dDT0wOz2dwQeeNCk7ZSqRQejydnqtnv9+Pf//3f8d///d/42Mc+hn/4h39ouCE7HyjnO/jzn/8cN998M9auXUsfR74j3G48InHcCN2kCiEUcs82nCNtZ0vgdDrx2GOP4bHHHgPLstQMvru7G9FolLZYymQyugEUS69we/2HhobqrhHEsmzOoJpKpaIFVovFApfLVbZkQz1AagdEGyeTyeDRRx/Fm9/8Zhw9ehS/+93vcMstt+AjH/lI08i+mJ81QSKRwIc+9CEcOHAAOp0Ou3fvxtDQEIDFXPyPf/xjiMVifPe738WOHTua8AwaBoH0zzacI21nRcGyLCwWC3bv3o1f/epXaG9vp1PAXV1dOTIQSqWSykCIxWIEAgHMzMw0bKCq2Pq9Xi8mJycRDodpN06hE0C9wS0Ua7VaDA0Noa2tDW63G1/+8pexZ88eBAIBvPOd78T1/7+9Ow9r6kr/AP4NUB2wiiCbU62ymKJssrrWUiyiFhAtyjIuSN1mXFo6YF3qVhFx1IqotH2Kih2rOFgBK9ZiELUilGERRhCjAsoisq8GBfL+/uDJ/ZEqCrKE5XyeJ0+T3HtzT2h8k3vOe97j6gpra2uZlDZ+1XrWEkFBQUhPT8d3332H0NBQhIeH48yZM8jMzISbmxsSExNRWFiIjz76CEKhUCZfrt2kwx9o1qffwzx58oQbUNLS0uKKiLUmMTERz58/h66uLvfc5s2bYWxsDC8vL24Mobfg8XgYOXIkvL29ERcXh6CgIJSUlGDOnDlYsGABoqKioKmpyfWvV1ZW4ubNm7h27Rru3r2Ld999F+bm5h1akvBN1dfXQygU4v79+xg5ciSsra3B5/NRUVGBP/74AxkZGSgrK+NmtXYVsViMgoICJCYmoq6uDmZmZuDz+aiursa2bdvg4OAAKysrpKam4uHDh3B1dcWvv/7apW16lcTEROjp6UFHRwcDBgyAq6srIiMjpfaJjIzEkiVLAADOzs6IiYkBESEyMhKurq4YOHAgtLW1oaenh8TERFm8jV6DZe/IAEs7axtJF9aWLVvw1Vdf4fbt2wgNDYWtrS10dXWhq6sLgUCA/fv3Y9SoUaiurkZWVhaUlZW5OkDdEfgl5Y1ramowatQo8Pl87rxDhw7F0KFDuTo2T548gVAohLKyMjQ1NTutVhEgnRWkpqbGZQWVlJTg0KFDuHz5MlavXo1du3ZJDR7b2NjAxsamU9rwJlpbz7q1fRQUFKCsrIyysjIUFBRIZeD0xjWvuxsL+jLA0s7aj8fjwcjICPr6+uDz+dizZw/y8vKgoKCAY8eOwdnZWeqXdVFREe7evQsVFRVoaWm9Uf7669TV1SEnJwcikQijR4+Gvr5+q+fg8XhQUVGBiooKt36u5AtAUsr4Tb+kJME+Ly9PatHz4uJiBAYGIiYmBmvXroWfn5/M6jkxPQcL+j2MJO1sw4YNb5x2JikC1xfTzoqKipCVlYXY2FiuIFlcXBxCQ0OxefNmTJo0SWox+PLychQUFODOnTsYNmwYtLS0OlzDpqPljXk8HlRVVaGqqsqVMm75JSWpZf+615R04+Tn50NDQ4Obd1BcXIyDBw8iNjYW69atw549e9qV9dTd3nnnHeTl5XGPXzZPRbLPiBEj0NjYiKqqKgwbNqxNx74OEcl0bd7uxgZye5jOSDvLzs7G48ePMWDAAHh7e2Pbtm1S5+irmRCNjY2IjY3F6dOnkZiYCGtrazg7O8PCwoIbYJWUgWhZB6iturq8sVgs5q4AqqqqWr1KaWpqQkFBAQoKCqTWGCgqKsLBgwdx/fp1fPbZZ/jb3/7Wo4O9xKvWs5Y4cuQI/ve//3EDuefOncN//vMfZGRkwN3dnRvInT59Ou7du9emgVxJocReNujLsncYaSwTotmzZ8+4xeBv377NLQZvaGjIlR0oKirCs2fPoK6uDi0trVZLDFRUVHR7eeM/59irqqpCXV0dNTU1KCwshJaWFkaOHAkFBQUUFRXhwIEDuHHjBry8vODu7t4t8wA608vWs966dSssLCzg6OiI+vp6LFq0CKmpqVBVVUVoaCh0dHQANI+FHTt2DAoKCggICMCsWbNeea7s7GzuWKB5AP5f//oXzM3NMXPmzJ7+eWdBn5H25wqdu3fvBgBs3LiR28fOzg7bt2/HpEmT0NjYCC0tLZSUlMDf319q35b79WZPnz7FL7/8gjNnziA3Nxcff/wx5s2bBz6fzy1lWFRUhMbGRmhoaHDrAUtWhxowYAC0tbUxePBgmbS/oaEBQqGQWwympqYGgwcPhp6eHgICAhAfHw8vLy+4ubnJLNi/Ls/+m2++QXBwMBQUFKCuro5jx45h1KhRAFpfY6Ir5OfnY+3atTh37hx4PB7Cw8O5z7iPj49UFlwPxQquMdJYJsSLlJSU4OLiAhcXF1RVVSE8PBybNm1CaWkp5syZA2dnZ5iamqKhoQFFRUVITU2FSCTC22+/jTFjxsisXHBTUxPy8vK4cZpp06aBx+MhIiIC27Ztg1AoxPvvv4/vvvsOlpaWMuuXbmpqwurVq6WuLh0dHaWuLk1NTZGUlAQlJSV8++23WL9+Pc6cOQMAUFRUxK1bt7q0jWKxGHJychgxYgTCw8O5KrD5+flYvnw51qxZg8bGRlRXV+Ptt9+WyXyF7sKCPtOvKCsrw8PDAx4eHigtLcXZs2fxj3/8A/X19dDX10dcXBz27t0LS0tLVFVV4f79+9xSi5qamt3SR96yMNxf//pXWFlZQV5eHgUFBdi3bx+Sk5Pxz3/+Ew4ODhAIBAgICICdnR2Xx97dWubZA+Dy7FsG/ZYrzk2cOBEnT57s8nZJBmiJCHJycqivr0d9fT1KSkowbdo0PH/+HGpqajh48CAEAgEUFRVRW1sLNzc3uLu7d3n7ZIUF/T5G1pkQvYmamhqWLVsGRUVF7N+/Hzk5OdDS0kJAQADmzZuHOXPmwNLSkltpKzU1FW+99RZXBqKzu1JaBvt33nmHC/Z5eXnYt28fbt26BW9vbwQFBXH9zk5OTnBycurUdrRXW64uWzp69KhUv3t9fT0sLCyk1pjoDJIrH8l/58yZgxUrVuCTTz6BnZ0dduzYgZ07d8LIyAjDhw9HXV0dfvnlFzx+/LhPZ/T03WuYfsrS0hL37t3jUgpDQ0Ph6OgotU/LaoRnz56FjY0NeDweHB0dERoaimfPniEnJwf37t2DlZWVLN5Gt2loaEBubi4EAgFiY2Nx9epVBAcHo6amBs7Ozvjkk08QHh4OVVVVWFlZgc/nQyQSISkpCWlpaXjy5Embq2y+qg3Z2dn473//CwUFBVhZWeHdd99FQUEB1q1bh0WLFmHGjBlISEiAi4tLTx9ofKWTJ08iKSkJPj4+3HMPHz5EUlISTp06hc8//xwPHjx4o9eWZOO05O/vj7CwMJSVlcHAwADq6uoAgH379uHQoUNobGyEoaEhlJSUcOPGDRw9ehQjRozoswEfQPMfqofeGCJqampq9zFRUVE0ZswY0tHRIV9fXyIi2rJlC0VGRhIRkUgkImdnZ9LV1SVLS0t68OABd6yvry/p6OgQn8+nixcvEhHRr7/+Snw+n3R1dWn37t0vnG///v00duxYMjIyIhsbG8rNzeW2ycnJkYmJCZmYmJCDg0O734ssicViyszMpK1bt5KpqSk5OTnRyZMnqaSkhGpra6moqIjS0tIoJiaGEhISKDc3l2pqaqiurq5Nt8rKSkpPTyeBQEBZWVncsZmZmeTp6UmWlpZ09uzZN/oMdKebN2/SjBkzuMd+fn7k5+f3wn6XL18mfX19evLkSauvtWTJEgoLC2t3G1r+jUQiEYlEIiIiCgkJobVr15K1tTVNmTKFGhoauP1mzpxJmzdvJiIiLy8vcnBwoMTExHafu5t1OLbKOrCzoP8aMTExZG9vT0+fPpXJ+RsbG0lHR4cePHhAz549I2NjY8rIyJDa58qVK1RXV0dEREFBQbRgwQJu26BBg7q1vV2lqamJUlJSyMfHh4yMjMjFxYXCwsKovLycamtrqbCwkFJTU0kgEFBiYiI9evSo1S+AyspKSktLI4FAQHfv3uX2y8jIIA8PD7KysqKff/65xwd7iYaGBtLW1qbs7GzuM3L79m2pfVJSUkhHR4eEQqHU8+Xl5VRfX09ERCUlJaSnp/fC56utKioq6PPPP6epU6fSF198wX0miYi2bdtGPB6Pdu7cSdevXyciotTUVJKTkyOxWEyVlZVvdE4ZYEG/L5P8o7ezsyOBQEBERMXFxXTmzBm6d+9et7Shrb/iJFJSUmjy5Mnc474S9FtqamqiuLg4Wrt2LRkYGJCHhwdduHCBqqqqqLa2lvLz8yk5OZkEAgElJSVRfn4+1dbWUkVFBXdlIBQKuWCfnp5OixcvpokTJ1J4eHivCfYtve7qcvr06aShofHCVV9cXBwZGhqSsbExGRoaUnBw8CvPIxaLKTU1lTuHxLNnz2ju3Lnk7+9PRESWlpbk7e1NZWVlREQUGxtL9vb2FBISQqamptwVa8ur3F6CBf3+wNPTky5dukQNDQ1kbW1Ny5cvp5SUFCJq/kfQlcLCwujTTz/lHv/444+0evXqVvdfvXo17dy5k3ssLy9P5ubmNGHCBAoPD+/KpspEY2MjxcTE0PLly2ncuHG0cuVKunz5MlVXV1NNTQ09evSIEhISKCoqiqKioiglJYWqqqqorq6O0tLSaOHChTRp0iSKjIyUabB/XRfe8ePHSU1NjQvaP/zwA7ctJCSE9PT0SE9Pj0JCQrq8rRcuXKB58+ZxjyV/t7KyMsrMzCQnJyeysLCgqVOnUlxcHBER7dy5kwICAoiI6MaNG5Sdnd3l7ewiHY6tLHunB2tqaoK8vDzmzJmDLVu2ICIiAubm5lJF1CQDTkSyn1IuGaS7du0a99zLFoLvBRNg2kxeXp6rUvn8+XMIBAKcOHECXl5emDx5MioqKsDj8eDr64u33noL58+fh7OzM9TV1SESieDr64uPP/5YpnnhbcmzBwAXFxccPnxY6rny8nLs2LEDSUlJ4PF4MDc3h6OjY6fObSgtLYW7uzv8/PxgYWGBW7duwczMDEDzzGtJETlVVVVs3LgRdnZ2WLVqFebPn4+goCCYmJjg+fPn3CDvlClTOq1tvREL+j2YJBBUVVWhtLQUEyZMgIeHB7ddJBIhLS0No0aNwvDhw6UCPlHnpJy1NY1TIBBg165duHbtGgYOHCh1PCC9EHxfCvotDRgwALNnz4aZmRl2796NCxcuQFdXF8XFxTh69CisrKyQkJCA4cOH44MPPoBQKMSOHTtQV1cHV1dXmbW7LXn2rfntt99ga2sLVVVVAICtrS0uXboENze3TmufmpoabG1tERgYCEVFRaioqGDIkCEAwH3WJGUy6uvruXIagwYNQnZ2NvLy8vDll19i0KBBndam3oylbPZgkqD95MkTfPbZZ5g7d67U9traWqSmpsLe3h6zZs1CcnLyC8d2VFtSQFNTU7Fy5UqcP39eqhR0awvB93UXL17E5MmTIRQKER0djevXr8PQ0BAbN26Eu7s7rl+/jl27diEsLAyxsbGYMGGCTNv7sjz7l83E/vnnn2FsbAxnZ2fuh0Bbj+0oHx8ffPvtt1BVVcUPP/wAOTk5VFZWoqamBlOmTMH69etRXV2Njz76CMHBwRg3bhw0NTXx73//G/r6+izgt8CCfg9F1Fx66P79+yguLoa8vDy+/vpruLu7Izo6GkBzeYHp06cjOTkZjo6OSE9PBwBERETg5s2bUq8DAMXFxfj999/btZqWgoICDh8+DDs7O4wdOxYLFiyAgYEBtm7dytVI8fHxQW1tLebPn4/x48dzXwp37tyBhYUFTExM8OGHH2LDhg149OgR3nvvPejp6XG1floKCQmBuro6xo8fj/HjxyM4OJjbduLECYwZMwZjxozh5hn0RJ6enlL59EOGDMHChQtx9+5d2NvbS3XlDB48GNra2rJqaps5ODggNzcX6enpsLW17fbZv2KxGIMGDcJXX30FQ0ND/P7773Bzc0NBQQFWrVqFsLAwGBkZYdGiRfD19UV4eDj27NnD1fdhWuiMgYEuujFEdPr0afLw8KDbt2+TWCymvXv30okTJ0goFNKnn35KH374IZmYmJCNjQ0tXbqUiIgSExNpxowZVFdXRyKRiC5fvkxERNXV1RQVFUWZmZlE9GZzADqiLemfx48ff+lAcVlZGWlra1NZWRmVl5eTtrY2lZeXd1fT+7T2Zmg1NjbSkCFDiIjo1KlTtGLFCm7bihUr6NSpU13WVpFIREZGRlRcXEybNm2iefPmka+vL9XW1nZ5UkMP0eHYyn7p93CzZs3C8uXLoa2tDR6PB29vbyxevBh+fn5QV1fHlStX8Ntvv0FVVRV8Ph8AoKuri4EDByI+Ph6zZ8/G8ePHQUSIj4+HmZkZxo4dC+D/xww6OqO0rdqyFmprWvYdq6iocH3HTMe1pQvv8ePH3P3z589znyE7OztER0ejoqICFRUViI6O7rI1GIgI1dXV4PP5UFJSwq5du7B9+3bU1NRALBb37Vm0nYgF/R5OWVkZkydPhpKSEpehAwDTpk1DYWEhcnNzcfbsWRQXF2Pq1KkAmgewRCIRfHx8sHTpUvz000/IycnBTz/9xJVcjomJ4YJmd2X89Ia+4/6oLV14gYGBMDAwgImJCQIDAxESEgKgOWNmy5YtsLS0hKWlJbZu3coN6nY2Ho+HmpoaCIVCLmPHyMgI/v7+Mit73Rux7J1epOVC6Q4ODkhISMC6deugpKQEsViMqVOnoqGhAQsXLsTTp09hZmaG+fPnAwBSUlIwdOhQODg44MCBA7h58ybq6uqwdetWeHt7Y8GCBbJ8axwHBwe4ublh4MCB+P7777FkyRJcuXJF1s3q0V5Xy97LywuxsbEAmtcWKC4uRmVlJQDpWvb6+vpStey//vpr7v7u3bu5tRn+zNPTE56enp35llpVU1ODFStWoKmpCQoKCuzX/RtgQb+XUlNTw/fffw+gebD38ePHSEtLg5eXF6ZMmYJDhw7BwcEBf/nLX0BEyMrKgoaGBlRVVXHz5k0YGBhg+/btyMrK4gZ7qZPSPFvTlvTPYcOGcfeXLVuG9evXc8devXpV6lhra+sua2tv0ZYc+wMHDnD3Dx06hNTUVO5xd9Sy70wmJiYYP368rJvRu3XGwEAX3Zg3kJOTQ42NjURE5OTkRNHR0XTv3j364osvKDQ0lIiIMjIyaNmyZXTkyJFubVtbarQUFhZy98+dO0cTJkwgouaB3NGjR1N5eTmVl5fT6NGjuSn2/Vl7B2EnTZpE0dHR3OO+WCajj2MDuYy00aNHc33069atw8iRI3H//n2UlpbC1NQU1dXVGDduHL755hucP38eO3fu7La2dUXf8aVLl16ZAurl5cWlf/L5fAwdOpTbJi8vz23788Blb9GesY6HDx8iJycHNjY23HOSWvYTJ05EREREVzeX6Qk645uji25MJxGLxZSVlUVERG5ubuTk5ESJiYlkZ2dHe/fu5fbpbdqSAtpSYGAgl9ZK1Dd+5banNpK/vz+tWbNG6rn8/Hwiai48NmrUKLp//37XNZbpDOyXPvN6PB4P7733HoDmVYvmzp2LXbt24f3338ff//53bp/epr0poKdPn+7U8gA9QXtWOwsNDX3h/b+sTAbTt7Gg388oKipi8eLFiIiIwKZNm3r19HTWtdG2HHsAyMrKQkVFBSZNmsQ911/LZPR3LHunH+uNv+7fVGhoKJydnaXmJPSFCqAtx0mamprg6enJjZNYWFhwXwChoaFwdXWV+n9+584drFy5EnJychCLxdiwYQML+v0Aj4hev5ds9NiGMT1DfHw8tm/fzk04k+SRb9y48YV9TU1NceTIEUyePPmlr+Xh4QF7e3s4Ozt3XYMZpuM6/EuNde8wvVZf6Nrw9PSEhoYGDA0NX7qdiLBu3Tro6enB2NgYKSkp3LbeUoCO6WE6YzS4i24M81qvW6aPqHl91C+//FLquNaW6Vu6dCmpq6uTgYHBS88nFotp7dq1pKurS0ZGRpScnMxte5MVpK5du0bJycmtni8qKopmzpxJYrGY4uPjycrKiohYAbp+rMOxVdaBnQV9pkeRRRDOyclp9Xx/rlrJ5/OpsLCw26tbMj0GS9lkmM40bdq0VxYMi4yMxOLFi8Hj8TBx4kRUVlbi8ePHXVYFtLUMJVaAjnlTLOgzTDuwIMz0dizoM0wP1trkq/ZMymKYlljQZ5h26O4g7OjoiB9//BFEhISEBCgrK2P48OHdungJ07ewyVkM0w6Ojo44fPgwXF1d8ccff0gF4U2bNqGiogIAEB0d3Wr9+Zbc3Nxw9epVlJaWYsSIEdixYwcaGhoAAKtWrcLs2bNx8eJF6OnpQUlJCcePHwcgXYAOQJcuXsL0LWxyFsO00DIIa2pqvhCEiQhr1qzBpUuXuCBsYWEBADh27Bj8/PwAAJs3b8bSpUtl9j6YPqvDk7NY0GcYhuk92IxchmEYpu1Y0GcYhulHWNBnGIbpR1jQZxiG6UdY0GcYhulHWNBnGIbpR1jQZxiG6UdY0GcYhulHWNBnGIbpR1jQZxiG6UdY0GcYhulHenKVzQ7XmGAYhmGksV/6DMMw/QgL+gzDMP0IC/oMwzD9CAv6DMMw/QgL+gzDMP0IC/oMwzD9yP8BembvSYMICUEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.array([[1, 0, 0],\n",
    "              [-3, 1, 0],\n",
    "              [7, 9, 1]])\n",
    "\n",
    "eigenvalues, eigenvectors = eig(A)\n",
    "eigenvectors = eigenvectors / np.linalg.norm(eigenvectors, axis=0)\n",
    "fig = plt.figure(figsize=(6,6))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "origin = [0, 0, 0]\n",
    "for i in range(len(eigenvalues)):\n",
    "    ax.quiver(*origin, *eigenvectors[:, i], color='r')\n",
    "\n",
    "ax.set_xlim([-1, 1])\n",
    "ax.set_ylim([-1, 1])\n",
    "ax.set_zlim([-1, 1])\n",
    "ax.set_xlabel('X-axis')\n",
    "ax.set_ylabel('Y-axis')\n",
    "ax.set_zlabel('Z-axis')\n",
    "ax.view_init(elev=20, azim=40)  # Adjust the view for better visualization\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e73c1df6",
   "metadata": {},
   "source": [
    "------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d4d7d639",
   "metadata": {},
   "source": [
    "<b> Question 6</b> \n",
    "\n",
    "Determine the eigenvalues and eigenvectors of the following matrices and write down a set of basis vectors for the eigenspace $E_{\\lambda}$\n",
    "\n",
    "(a)$A = \\begin{pmatrix}\n",
    "5&0&0\\\\\n",
    "0&5&0\\\\\n",
    "0&0&2\\\\\n",
    "\\end{pmatrix}\\qquad$\n",
    "(b)$B = \\begin{pmatrix}\n",
    "1&2&3\\\\\n",
    "0&5&1\\\\\n",
    "0&0&9\\\\\n",
    "\\end{pmatrix}\\qquad$\n",
    "(c)$C = \\begin{pmatrix}\n",
    "-2&0&0\\\\\n",
    "2&-2&0\\\\\n",
    "4&10&-2\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53fd8556",
   "metadata": {},
   "source": [
    "<b>Solution (a)</b>\n",
    "\n",
    "We know that, <br>\n",
    "If a $n \\times n$ matrix $A$ is diagonal or triangular matrix then the eigenvalues of $A$ are the entries along the leading diagonal\n",
    "\n",
    "Therefore, eigevalues for $A$ are $5, 5$, and $2$ i.e, $\\lambda_{1,2} = 5$ and $\\lambda_{3} = 2$\n",
    "\n",
    "For eigenvector, when $\\lambda = 5$<br>\n",
    "we have to solve $(A - \\lambda I)u - 0$<br>\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "5-5&&0\\\\\n",
    "0&5-5&0\\\\\n",
    "0&0&2-5\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "0&0&0\\\\\n",
    "0&0&0\\\\\n",
    "0&0&-3\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -3z = 0 \\Rightarrow z =0$<br>\n",
    "\n",
    "We have 1 non-zero equation & 3 unknown, therefore free variables $x,y$ can  take any number $s$ and $t$.\n",
    "\n",
    "$\\Rightarrow u = \\begin{pmatrix}\n",
    "s\\\\\n",
    "t\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$ or $s\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix} + t\\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Thus, basis vector for $E_{5} = \\{\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}\\}$\n",
    "\n",
    "\n",
    "For eigenvector, when $\\lambda = 2$<br>\n",
    "we have to solve $(A - \\lambda I)u - 0$<br>\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "5-2&&0\\\\\n",
    "0&5-2&0\\\\\n",
    "0&0&2-2\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "3&0&0\\\\\n",
    "0&3&0\\\\\n",
    "0&0&0\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow 3x = 0$<br>\n",
    "and, $\\Rightarrow 3y = 0$\n",
    "\n",
    "$\\Rightarrow x = y= 0$ and $z$ can be any non-zero number, $s$.\n",
    "\n",
    "Therefore, eigenvector for $E_{2}$ is $u = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "s\\\\\n",
    "\\end{pmatrix}$ or $s\\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$. This basis vector is $\\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "<b>Solution (b)</b>\n",
    "\n",
    "We know that, <br>\n",
    "If a $n \\times n$ matrix $A$ is diagonal or triangular matrix then the eigenvalues of $A$ are the entries along the leading diagonal\n",
    "\n",
    "Therefore, eigevalues for $A$ are $1, 5$, and $9$ i.e, $\\lambda_{1} = 1, \\lambda_{2} = 5$ and $\\lambda_{3} = 9$\n",
    "\n",
    "For eigenvector, when $\\lambda = 1$<br>\n",
    "we have to solve $(A - \\lambda I)u - 0$<br>\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1-1&2&3\\\\\n",
    "0&5-1&1\\\\\n",
    "0&0&9-1\\\\\n",
    "\\end{pmatrix}\\cdot\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "0&2&3\\\\\n",
    "0&4&1\\\\\n",
    "0&0&8\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow 2y + 3z =0$<br>\n",
    "and, $4y + z = 0$<br>\n",
    "and $8z = 0$\n",
    "\n",
    "$\\Rightarrow z = 0$ and $y =0$. Therefore $x$ can take any non-zero $s$, making eigenvector for $\\lambda_{1} = s\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "For eigenvector, when $\\lambda = 5$<br>\n",
    "we have to solve $(A - \\lambda I)u - 0$<br>\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1-5&2&3\\\\\n",
    "0&5-5&1\\\\\n",
    "0&0&9-5\\\\\n",
    "\\end{pmatrix}\\cdot\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-4&2&3\\\\\n",
    "0&0&1\\\\\n",
    "0&0&3\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -4x + 2y + 3z = 0$<br>\n",
    "and, $z = 0$<br>\n",
    "and $3z = 0$\n",
    "\n",
    "$\\Rightarrow z = 0$ and $y =2x$. Therefore if $x$ can take any non-zero $s$, making eigenvector for $\\lambda_{1} = s\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "For eigenvector, when $\\lambda = 9$<br>\n",
    "we have to solve $(A - \\lambda I)u - 0$<br>\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1-9&2&3\\\\\n",
    "0&5-9&1\\\\\n",
    "0&0&9-9\\\\\n",
    "\\end{pmatrix}\\cdot\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-8&2&3\\\\\n",
    "0&-4&1\\\\\n",
    "0&0&0\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -8x + 2y + 3z = 0$<br>\n",
    "and, $-4y + z = 0$<br>\n",
    "\n",
    "\n",
    "$\\Rightarrow z = 4y$ and solving first equation, we get <br>\n",
    "$-8x + 2s +3(4s) = 0 \\Rightarrow 8x = 14s$\n",
    "\n",
    "$x = s \\frac{14}{8} = s\\frac{7}{4}$\n",
    "\n",
    "Therefore, eigenvector $u = \\begin{pmatrix}\n",
    "7/4\\\\\n",
    "1\\\\\n",
    "4\\\\\n",
    "\\end{pmatrix}$ or $4\\begin{pmatrix}\n",
    "7\\\\\n",
    "4\\\\\n",
    "16\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Thus, basis vector for eigenspaces $E_{1},E_{2}$ and $E_{3}$ are $\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix},\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$, and $\\begin{pmatrix}\n",
    "7\\\\\n",
    "4\\\\\n",
    "16\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "<b>Solution (c)</b>\n",
    "\n",
    "We know that, <br>\n",
    "If a $n \\times n$ matrix $A$ is diagonal or triangular matrix then the eigenvalues of $A$ are the entries along the leading diagonal\n",
    "\n",
    "Therefore, eigevalues for $A$ is $-2$ i.e, $\\lambda_{1,2,3} = -2$\n",
    "\n",
    "For eigenvector, when $\\lambda = -2$<br>\n",
    "we have to solve $(A - \\lambda I)u - 0$<br>\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-2+2&0&0\\\\\n",
    "2&-2+2&0\\\\\n",
    "4&10&-2+2\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "0&0&0\\\\\n",
    "2&0&0\\\\\n",
    "4&10&0\\\\\n",
    "\\end{pmatrix}\\cdot\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow 2x = 0$ or $x = 0$<br>\n",
    "and, solving last equation, we get $y =0$<br>\n",
    "This means $z$ can be any non-zero $s$.\n",
    "\n",
    "Therefore, eigenvector is $u = s\\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$ and basis vector is $\\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aeaaa314",
   "metadata": {},
   "source": [
    "-----------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de500f7f",
   "metadata": {},
   "source": [
    "<b>Question 7</b>\n",
    "\n",
    "Determine the eigenvalues and eigenvectors of the following matrices :\n",
    "\n",
    "(a)$A = \\begin{pmatrix}\n",
    "7&0&2&3\\\\\n",
    "0&7&4&6\\\\\n",
    "0&0&5&-3\\\\\n",
    "0&0&0&5\\\\\n",
    "\\end{pmatrix}\\qquad$\n",
    "(b)$B = \\begin{pmatrix}\n",
    "1&0&0&0\\\\\n",
    "0&1&0&0\\\\\n",
    "0&0&1&0\\\\\n",
    "0&0&0&3\\\\\n",
    "\\end{pmatrix}\\qquad$\n",
    "(c)$C = \\begin{pmatrix}\n",
    "3&0&0&0\\\\\n",
    "0&3&0&0\\\\\n",
    "0&0&3&0\\\\\n",
    "0&0&0&3\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43bba723",
   "metadata": {},
   "source": [
    "<b>Solution (a) </b>\n",
    "\n",
    "Since, $A$ is an upper triangular matrix, eigenvalues are $\\lambda_{1,2} = 7$ and $\\lambda_{3,4} = 5$\n",
    "\n",
    "Next, for eigenvector when $\\lambda_{1,2} = 7$<br>\n",
    "with $(A - \\lambda I)u = 0$<br>\n",
    "\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "7-7&0&2&3\\\\\n",
    "0&7-7&4&6\\\\\n",
    "0&0&5-7&3\\\\\n",
    "0&0&0&5-7\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "w\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "0&0&2&3\\\\\n",
    "0&0&4&6\\\\\n",
    "0&0&-2&3\\\\\n",
    "0&0&0&-2\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "w\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -2w = 0$ or $w = 0$<br>\n",
    "$\\Rightarrow -2z - 3w = 0$ or $z = 0$. Then $x, y$ can be any non-zero $s,t$\n",
    "\n",
    "Thus, eigenvector, $u = s\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix} + t\\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Similarly, for eigenvector when $\\lambda_{3,4} = 5$<br>\n",
    "with $(A - \\lambda I)u = 0$<br>\n",
    "\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "7-5&0&2&3\\\\\n",
    "0&7-5&4&6\\\\\n",
    "0&0&5-5&3\\\\\n",
    "0&0&0&5-5\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "w\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "2&0&2&3\\\\\n",
    "0&2&4&6\\\\\n",
    "0&0&0&3\\\\\n",
    "0&0&0&0\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "w\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow w = 0$<br>\n",
    "and, from the second row, $y + 2z + 6w = 0$ or $y + 2z  = 0$<br>\n",
    "we get, $y = -2z$\n",
    "\n",
    "If $z = s$, then $y = -2s$, <br>\n",
    "and, from the first row $2x + 2z +3w = 0$ or $x = -z$<br>\n",
    "$x = -s$\n",
    "\n",
    "Thus, eigenvector $ u = \\begin{pmatrix}\n",
    "-s\\\\\n",
    "-2s\\\\\n",
    "s\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$ or $u = s\\begin{pmatrix}\n",
    "-1\\\\\n",
    "-2\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "<b>Solution (c)</b>\n",
    "\n",
    "Since, $C$ is a diagonal matrix, eigenvalues are $\\lambda_{1,2,3,4} = 3$\n",
    "\n",
    "Then, for eigenvectors <br>\n",
    "with $(A - \\lambda I)u = 0$<br>\n",
    "\n",
    "we get, $\\begin{pmatrix}\n",
    "3-3&0&0&0\\\\\n",
    "0&3-3&0&0\\\\\n",
    "0&0&3-3&0\\\\\n",
    "0&0&0&3-3\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "w\\\\\n",
    "\\end{pmatrix}\n",
    "= \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "0&0&0&0\\\\\n",
    "0&0&0&0\\\\\n",
    "0&0&0&0\\\\\n",
    "0&0&0&0\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "w\\\\\n",
    "\\end{pmatrix}\n",
    "= \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Since, all rows are zero, then all $x, y, z, w$ are free variables, and can take any non-zero $s, t, a, b$\n",
    "\n",
    "Therefore, $u$ for $\\lambda_{1,2,3,4}$ = $\\begin{pmatrix}\n",
    "s\\\\\n",
    "t\\\\\n",
    "a\\\\\n",
    "b\\\\\n",
    "\\end{pmatrix}$ or $u = s\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix} + t\\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix} + a\\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix} + b\\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9735922a",
   "metadata": {},
   "source": [
    "----------------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "140e95e7",
   "metadata": {},
   "source": [
    "<b>Question 8</b> \n",
    "\n",
    "Prove proposition 7.7 which says a matrix is invertible $\\iff \\lambda = 0$ is not an eigenvalue"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "976478d4",
   "metadata": {},
   "source": [
    "<b>Solution</b>\n",
    "\n",
    "We know that, a matrix $A$ is invertible $\\iff det(A) = 0$\n",
    "\n",
    "Here, we have $\\lambda = 0$.\n",
    "\n",
    "Then, the characteristic equation $det(A - \\lambda I) = 0$<br>\n",
    "$\\Rightarrow det(A - 0 I) = 0$<br>\n",
    "$\\Rightarrow det(A) = 0$\n",
    "\n",
    "therefore, matrix $A$ is not invertible. Thus, $\\lambda = 0$ is not an eigenvalue of a invertible matrix\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6364e727",
   "metadata": {},
   "source": [
    "--------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eef83990",
   "metadata": {},
   "source": [
    "<b>Question 9</b>\n",
    "\n",
    "Let $A$ be a square matrix and $\\lambda$ be an eigenvalue with the corresponding eigenvector $u$. Prove that the eigenvalue $\\lambda$ is unique for the eigenvector $u$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a24649f",
   "metadata": {},
   "source": [
    "<b>Solution</b> \n",
    "\n",
    "If there is an eigenvector $u$ belonging to two distinct $\\lambda$ and $t$.\n",
    "\n",
    "Then, <br>\n",
    "$\\qquad Au = \\lambda u$<br>\n",
    "$\\qquad Au = tu$\n",
    "\n",
    "But, $Au - Au = 0$<br>\n",
    "$\\Rightarrow \\lambda u - tu = 0$<br>\n",
    "\n",
    "$\\Rightarrow (\\lambda - t)u = 0$\n",
    "\n",
    "Again, $u$, an eigenvector cannot be zero.Therefore, <br>\n",
    "$\\qquad \\lambda - t = 0$<br>\n",
    "$\\qquad \\lambda = t$<br>\n",
    "\n",
    "Therfore, our supposition is wrong, $\\lambda$ and $t$ are not two distinct eigenvalues, but same and unique"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5cc440b",
   "metadata": {},
   "source": [
    "-----------------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d9e641e8",
   "metadata": {},
   "source": [
    "<b> Question 10 </b> \n",
    "\n",
    "Let A be a 2 by 2 matrix. By using the result of question 3, the characteristic polynomial $p_{\\lambda}$ is given by\n",
    "\n",
    "$p_{\\lambda} = \\lambda^{2} - tr(A)\\lambda + det(A)$\n",
    "\n",
    "Show that if $[tr(A)]^{2} \\gt 4det(A)$ then $A$ has distinct eigenvalues.\n",
    "\n",
    "State under what conditions we have equal and complex eigenvalues."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d467a1e3",
   "metadata": {},
   "source": [
    "<b> Solution</b> "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5c308e5",
   "metadata": {},
   "source": [
    "--------------------------- "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "54026ac5",
   "metadata": {},
   "source": [
    "<b>Question 11</b> \n",
    "\n",
    "For each of the following matrices:\n",
    "\n",
    "<b>(a)</b>$A = \\begin{pmatrix}\n",
    "1&0&0&0\\\\\n",
    "3&2&0&0\\\\\n",
    "5&2&3&0\\\\\n",
    "9&8&1&4\\\\\n",
    "\\end{pmatrix}\\quad$<b>(b)</b>$A = \\begin{pmatrix}\n",
    "-1&3&4&7\\\\\n",
    "0&6&-3&5\\\\\n",
    "0&0&-8&9\\\\\n",
    "0&0&0&3\\\\\n",
    "\\end{pmatrix}\\quad$<b>(c)</b>$A = \\begin{pmatrix}\n",
    "2&0&0&0\\\\\n",
    "0&-4&0&0\\\\\n",
    "0&0&-7&0\\\\\n",
    "0&0&0&0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Determine: \n",
    "(i)Eigenvalue of $A\\quad$(ii)Eigenvalue of $A^5\\quad$(iii)Eigenvalues of $A^{-1}\\quad$(iv)$det(A)\\quad$(v)$tr(A)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5873ecbc",
   "metadata": {},
   "source": [
    "<b> Solution (a)</b> \n",
    "\n",
    "Since, $A$ is lower triangular matrix.\n",
    "\n",
    "(i) Eigenvalues of $A$ are:\n",
    "\n",
    "$\\lambda_{1} = 1, \\lambda_{2} = 2, \\lambda_{3} = 3$ and $\\lambda_{4} = 4$\n",
    "\n",
    "(ii) Eigenvalues of $A^5$ are:\n",
    "\n",
    "$(\\lambda_{1})^5 = 1^5 = 1, (\\lambda_{2})^5 = 2^5 = 32, (\\lambda_{3})^5 = 3^5 = 243$ and $(\\lambda_{4})^5 = 1024$\n",
    "\n",
    "(iii)Eigenvalues of $A^{-1}$ are:\n",
    "\n",
    "$(\\lambda_{1})^{-1} = 1, (\\lambda_{2})^{-1} = 2^{-1} = \\frac{1}{2}, (\\lambda_{3})^{-1} = \\frac{1}{3}$ and $(\\lambda_{4})^{-1} = \\frac{1}{4}$\n",
    "\n",
    "(iv) $det(A) = 1 \\times 2 \\times 3 \\times 4 = 24$\n",
    "\n",
    "(v) $tr(A) = 1+2+3+4 = 10$\n",
    "\n",
    "<b>Solution(b)</b>\n",
    "\n",
    "Since, A is an upper triangular matrix.\n",
    "\n",
    "(i) Eigenvalues of $A$ are:\n",
    "\n",
    "$\\lambda_{1} = -1, \\lambda_{2} = 6, \\lambda_{3} = -8$ and $\\lambda_{4} = 3$\n",
    "\n",
    "(ii)Eigenvalues of $A^5$ are:\n",
    "\n",
    "$(\\lambda_{1})^5 = (-1)^5 = -1, (\\lambda_{2})^5 = 6^5 = 7776, (\\lambda_{3})^5 = (-8)^5 = -32768$ and $(\\lambda_{4})^5 = 3^5 = 243$\n",
    "\n",
    "(iii)Eigenvalues of $A^{-1}$ are:\n",
    "\n",
    "$(\\lambda_{1})^{-1} = -1, (\\lambda_{2})^{-1} = \\frac{1}{6}, (\\lambda_{3})^{-1} = \\frac{1}{8}$ and $(\\lambda_{4}^{-1} = \\frac{1}{3}$\n",
    "\n",
    "(iv)$det(A) = -1 \\times 6 \\times (-8) \\times 3 = 144$\n",
    "\n",
    "(v) $tr(A) = -1 + 6 -8 + 3 = 0$\n",
    "\n",
    "<b>Solution (c) </b>\n",
    "\n",
    "Since $A$ is a triangular matrix, lower or upper.\n",
    "\n",
    "(i) Eigenvalues of $A$ are:\n",
    "\n",
    "$\\lambda_{1} = 2, \\lambda_{2} = -4, \\lambda_{3} = -7$ and $\\lambda_{4} = 0$\n",
    "\n",
    "(ii)Eigenvalues of $A^5$ are:\n",
    "\n",
    "$(\\lambda_{1})^5 = (2)^5 = 32, (\\lambda_{2})^5 = (-4)^5 = -1024, (\\lambda_{3})^5 = (-7)^5 = -16807$ and $(\\lambda_{4})^5 = 0^5 = 0$\n",
    "\n",
    "(iii)Eigenvalues of $A^{-1}$ are: \n",
    "\n",
    "$A^{-1}$ is not invertible for $\\lambda_{4} = 0$\n",
    "\n",
    "(iv)$det(A) = 2 \\times (-4) \\times (-7) \\times 0 = 0$\n",
    "\n",
    "(v)$tr(A) = 2 - 4 -7 + 0 = -9$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6fed435c",
   "metadata": {},
   "source": [
    "------------------------------------------------ "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c6abec9",
   "metadata": {},
   "source": [
    "<b> Question 12</b>\n",
    "\n",
    "Find the characterisitc polynomial $p(\\lambda)$ of the matrix $A = \\begin{pmatrix}\n",
    "2&1\\\\\n",
    "-2&1\\\\\n",
    "\\end{pmatrix}$ and determine the inverse of this matrix by using the Cayley-Hamilton theorem. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43ec8fb6",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "Here, the characterisitc equation would be:\n",
    "\n",
    "$p(\\lambda) = det\\begin{pmatrix}\n",
    "2-\\lambda&1\\\\\n",
    "-2&1-\\lambda\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow (2- \\lambda)(1- \\lambda) -(-2)1$<br>\n",
    "$\\Rightarrow 2 - \\lambda - 2\\lambda + \\lambda^2 + 2$\n",
    "\n",
    "$\\Rightarrow p(\\lambda) = \\lambda^2 -3\\lambda + 4$\n",
    "\n",
    "Substituting matrix $A$ into this equation yields:\n",
    "\n",
    "$p(A) = A^2 - 3A + 4I = O$\n",
    "\n",
    "$\\Rightarrow A^2 - 3A = -4I$\n",
    "\n",
    "$\\Rightarrow A(A - 3I) = -4I$\n",
    "\n",
    "$\\Rightarrow A\\left[\\frac{1}{-4}(A - 3I)\\right] = I$\n",
    "\n",
    "We know that, <br>\n",
    "$AA^{-1} = I$\n",
    "\n",
    "Therefore, $A^{-1} = \\frac{1}{-4}(A - 3I)$\n",
    "\n",
    "$\\qquad = \\frac{1}{-4}\\left(\\begin{pmatrix}\n",
    "2&1\\\\\n",
    "-2&1\\\\\n",
    "\\end{pmatrix} -3\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}\\right)$\n",
    "\n",
    "$\\qquad = \\frac{1}{-4}\\begin{pmatrix}\n",
    "2-3&1\\\\\n",
    "-2&1-3\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\qquad = \\frac{1}{-4}\\begin{pmatrix}\n",
    "-1&1\\\\\n",
    "-2&-2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\qquad \\Rightarrow A^{-1}= \\frac{1}{4}\\begin{pmatrix}\n",
    "1&-1\\\\\n",
    "2&2\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c427168",
   "metadata": {},
   "source": [
    "------------------------------- "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "13ecce76",
   "metadata": {},
   "source": [
    "<b>Question 13</b> \n",
    "\n",
    "Let $A = \\begin{pmatrix}\n",
    "6&5\\\\\n",
    "3&4\\\\\n",
    "\\end{pmatrix}$. By using the Cayley-Hamilton theorem determine $A^2$ and $A^3$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c5db843",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "Here, the characteristic equation $p(\\lambda)$ would be :\n",
    "\n",
    "$\\qquad = det\\begin{pmatrix}\n",
    "6 - \\lambda&5\\\\\n",
    "3&4- \\lambda\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\qquad = (6 - \\lambda)(4 - \\lambda) - 3(5)$<br>\n",
    "$\\qquad = 24 -4\\lambda - 6\\lambda + \\lambda^2 + 15$<br>\n",
    "$\\qquad = \\lambda^2 - 10\\lambda + 9$\n",
    "\n",
    "Substituting matrix $A$ into this equation yields :\n",
    "\n",
    "$p(A) = A^2 - 10A - 9I = O$\n",
    "\n",
    "$\\Rightarrow A^2 = 10A - 9I$\n",
    "\n",
    "$\\qquad = 10\\begin{pmatrix}\n",
    "6&5\\\\\n",
    "3&4\\\\\n",
    "\\end{pmatrix} - 9\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\qquad = \\begin{pmatrix}\n",
    "60&50\\\\\n",
    "30&40\\\\\n",
    "\\end{pmatrix} - \\begin{pmatrix}\n",
    "9&0\\\\\n",
    "0&9\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow A^2 = \\begin{pmatrix}\n",
    "51&50\\\\\n",
    "30&31\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, for $A^3$, we can solve it as :\n",
    "\n",
    "$A^3 = A^2A$<br>\n",
    "$\\quad = A(10A - 9I)$\n",
    "\n",
    "$\\Rightarrow A^3 = 10A^2 - 9A$<br>\n",
    "and, using A^2 from above, we can get $A^3$ as:\n",
    "\n",
    "$A^3 = 10\\begin{pmatrix}\n",
    "51&50\\\\\n",
    "30&31\\\\\n",
    "\\end{pmatrix} - 9\\begin{pmatrix}\n",
    "6&5\\\\\n",
    "3&4\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow A^3 = \\begin{pmatrix}\n",
    "510&500\\\\\n",
    "300&310\\\\\n",
    "\\end{pmatrix} - 9\\begin{pmatrix}\n",
    "54&45\\\\\n",
    "27&36\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "$\\Rightarrow A^3 = \\begin{pmatrix}\n",
    "510-54&500-45\\\\\n",
    "300-27&310-36\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow A^3 = \\begin{pmatrix}\n",
    "456&455\\\\\n",
    "273&274\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81b38994",
   "metadata": {},
   "source": [
    "---------------------------- "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83ce9bc0",
   "metadata": {},
   "source": [
    "<b>Question 14</b> \n",
    "\n",
    "Let $A = \\begin{pmatrix}\n",
    "1&2&1\\\\\n",
    "2&1&1\\\\\n",
    "1&1&2\\\\\n",
    "\\end{pmatrix}$ and the characteristic polynomial of the matrix be given by $p(\\lambda) = \\lambda^3 - 4\\lambda^2 -\\lambda + 4$<br>\n",
    "Determine expressions for $A^{-1}$ and $A^4$ in terms of matrices $A, I$ and $A^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94525dfa",
   "metadata": {},
   "source": [
    "<b> Solution</b> \n",
    "\n",
    "Cayley-Hamilton theorem give us :\n",
    "\n",
    "$p(A) = A^3 - 4A^2 - A + 4I = o$\n",
    "\n",
    "$\\Rightarrow A^3 - 4A^2 - A = -4I$<br>\n",
    "$\\Rightarrow A(A^2 -4A - I) = -4I$<br>\n",
    "\n",
    "$\\Rightarrow A\\frac{1}{-4}(A^2 -4A - I) = I$\n",
    "\n",
    "Since, <br>\n",
    "$AA^{-1} = I$\n",
    "\n",
    "Therefore, $A^{-1} = \\frac{1}{-4}(A^2 -4A - I)$\n",
    "\n",
    "As the, Cayley theorem provides :\n",
    "\n",
    "$p(A) = A^3 - 4A^2 - A + 4I = o$\n",
    "\n",
    "$\\Rightarrow A^3 - 4A^2 - A = -4I$<br>\n",
    "\n",
    "$\\Rightarrow A^3 = 4A^2 + A -4I$<br>\n",
    "\n",
    "Now, we can get $A^4$ by multiplying $A^3$ by $A$\n",
    "\n",
    "$A^4 = A(A^3)$<br>\n",
    "$\\quad = A(4A^2 + A - 4I)$<br>\n",
    "$\\quad =4A^3 + A^2 -4A$<br>\n",
    "\n",
    "$\\quad = 4(4A^2 + A - 4I) + A^2 -4A$<br>\n",
    "$\\quad = 16A^2 + 4A - 16I + A^2 -4A$<br>\n",
    "$\\quad = 17A^2 - 16I$<br>\n",
    "\n",
    "Thus, $A^4 = 17A^2 - 16I$<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "454a9d70",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}