notebooks/Chapter 3 - Determinant.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy as sy\n",
    "sy.init_printing() \n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('ggplot')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.set_printoptions(precision=3)\n",
    "np.set_printoptions(suppress=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "def round_expr(expr, num_digits):\n",
    "    return expr.xreplace({n : round(n, num_digits) for n in expr.atoms(sy.Number)})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Visualization of Determinants</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The physical interpretation of determinants is to compute the area enclosed by vectors. For instance, consider the matrix\n",
    "$$\n",
    "A = \\left[\\begin{matrix} a & b \\\\ c & d \\end{matrix}\\right]\n",
    "$$\n",
    "The determinant represents the area of the parallelogram formed by the vectors\n",
    "$$\n",
    "\\left[\\begin{matrix} a \\\\ c \\end{matrix}\\right] \\quad \\text{and} \\quad \\left[\\begin{matrix} b \\\\ d \\end{matrix}\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we demonstrate with a matrix\n",
    "$$\n",
    "\\left[\\begin{matrix} 2 & 0\\cr 0 & 3 \\end{matrix}\\right]\n",
    "$$\n",
    "It's also easy to understand the area formed by these two vectors are actually a rectangle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "matrix = np.array([[2, 0], [0, 3]])\n",
    "def plot_2ddet(matrix):\n",
    "    # Calculate the determinant of the matrix\n",
    "    det = np.linalg.det(matrix)\n",
    "\n",
    "    # Create a figure and axis\n",
    "    fig, ax = plt.subplots(figsize=(6, 6))\n",
    "\n",
    "    # Plot the vectors\n",
    "    ax.arrow(0, 0, matrix[0,0], matrix[0,1], head_width=0.1, head_length=0.1, color='b')\n",
    "    ax.arrow(0, 0, matrix[1,0], matrix[1,1], head_width=0.1, head_length=0.1, color='r')\n",
    "    ax.arrow(matrix[0,0], matrix[0,1], matrix[1,0], matrix[1,1], head_width=0.1, head_length=0.1, color='r')\n",
    "    ax.arrow(matrix[1,0], matrix[1,1], matrix[0,0], matrix[0,1], head_width=0.1, head_length=0.1, color='b')\n",
    "\n",
    "    # Annotate the determinant value\n",
    "    ax.annotate(f'determinant = {det:.2f}', (matrix[0,0]/2, matrix[0,1]/2))\n",
    "\n",
    "    # Add labels and show the plot\n",
    "    ax.set_xlabel('x')\n",
    "    ax.set_ylabel('y')\n",
    "    ax.grid()\n",
    "    plt.show()\n",
    "    \n",
    "if __name__ == '__main__':\n",
    "    plot_2ddet(matrix)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the matrix is not diagonal, the area will take the form of a parallelogram. Similarly, in 3D space, the determinant is the volume of a parallelepiped. First let's draw a parallelogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "matrix = np.array([[2, 3], [1, 3]])\n",
    "plot_2ddet(matrix)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, determinant can be a _negative number_, it means we flip the area like flipping a piece of paper."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if two vectors are linearly dependent? The area between vectors will be zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "matrix = np.array([[2, 3], [4, 6]])\n",
    "plot_2ddet(matrix)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"background-color:Bisque; color:DarkBlue; padding:30px;\">\n",
    "This is the exact reason if we want a matrix to have full rank or linear independent columns, the determinant can't equal to zero!\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a plot of parallelepiped, in case you are not sure how it looks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "fig = plt.figure(figsize=(8, 8))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "# Define the vertices of the parallelepiped\n",
    "vertices = np.array([[2, 0, 0], [2, 3, 0], [0, 3, 0], [0, 0, 0],\n",
    "                     [2, 3, 4], [2, 6, 4], [0, 6, 4], [0, 3, 4]])\n",
    "\n",
    "# Plot edges between corresponding vertices\n",
    "def plot_edges(vertices, pairs, color):\n",
    "    for (i, j) in pairs:\n",
    "        ax.plot([vertices[i, 0], vertices[j, 0]],\n",
    "                [vertices[i, 1], vertices[j, 1]],\n",
    "                [vertices[i, 2], vertices[j, 2]], color=color)\n",
    "\n",
    "# Pairs of indices representing edges to be plotted\n",
    "bottom_edges = [(i, (i+1)%4) for i in range(4)]\n",
    "top_edges = [(i+4, (i+1)%4+4) for i in range(4)]\n",
    "vertical_edges = [(i, i+4) for i in range(4)]\n",
    "\n",
    "# Plot all edges\n",
    "plot_edges(vertices, bottom_edges, 'blue')\n",
    "plot_edges(vertices, top_edges, 'k')\n",
    "plot_edges(vertices, vertical_edges, 'red')\n",
    "\n",
    "# Add annotations for the coordinates at each vertex\n",
    "for vertex in vertices:\n",
    "    ax.text(vertex[0], vertex[1], vertex[2], f\"({vertex[0]}, {vertex[1]}, {vertex[2]})\")\n",
    "\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Computation of Determinants</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For $2\\times 2$ matrix $A$, the algorithm of determinant is\n",
    "$$\n",
    "  A=\\left|\\begin{matrix} a & b\\cr c & d \\end{matrix}\\right| \\qquad\\text{is equivalent to}\\qquad \\text{det}  A=ad-bc\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we experiment with SymPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c, d, e, f, g, h, i = sy.symbols('a, b, c, d, e, f, g, h, i', real = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With defined symbols, the algorithms of $2\\times 2$ and $3\\times 3$ determinants are"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle a d - b c$"
      ],
      "text/plain": [
       "a⋅d - b⋅c"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[a, b], [c, d]])\n",
    "A.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Anything above than $2 \\times 2$ is become more complicated"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle a e i - a f h - b d i + b f g + c d h - c e g$"
      ],
      "text/plain": [
       "a⋅e⋅i - a⋅f⋅h - b⋅d⋅i + b⋅f⋅g + c⋅d⋅h - c⋅e⋅g"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = sy.Matrix([[a, b, c], [d, e, f], [g, h, i]])\n",
    "B.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Cofactor Expansion</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $(i,j)$-_cofactor_ of $A$ is denoted as $C_{ij}$ and is given by\n",
    "$$C_{ij} = (-1)^{i+j} \\operatorname{det} A_{ij} = (-1)^{i+j} M_{ij}$$\n",
    "where $M_{ij}$ is the _minor determinant_ obtained by excluding the $i$-th row and $j$-th column.\n",
    "\n",
    "Let's learn from an example. Consider the matrix $A$:\n",
    "$$\n",
    "A = \\left[\\begin{array}{rrr}\n",
    "1 & 5 & 0 \\\\\n",
    "2 & 4 & -1 \\\\\n",
    "0 & -2 & 0\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "Any determinant can be expanded along an arbitrary row or column. We will expand the determinant along the first row:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\operatorname{det} A &= 1 \\cdot \\operatorname{det}\\left[\\begin{array}{rr}\n",
    "4 & -1 \\\\\n",
    "-2 & 0\n",
    "\\end{array}\\right] - 5 \\cdot \\operatorname{det}\\left[\\begin{array}{cc}\n",
    "2 & -1 \\\\\n",
    "0 & 0\n",
    "\\end{array}\\right] + 0 \\cdot \\operatorname{det}\\left[\\begin{array}{rr}\n",
    "2 & 4 \\\\\n",
    "0 & -2\n",
    "\\end{array}\\right] \\\\\n",
    "&= 1 \\cdot (0 - (-2)) - 5 \\cdot (0 - 0) + 0 \\cdot (-4 - 0) \\\\\n",
    "&= 1 \\cdot 2 - 5 \\cdot 0 + 0 \\cdot (-4) \\\\\n",
    "&= 2 - 0 + 0 \\\\\n",
    "&= 2\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The scalars $1$, $5$, and $0$ in front of each minor determinant are the elements of the first row of $A$.\n",
    "\n",
    "In general, the expansions across the $i$-th row or $j$-th column are:\n",
    "$$\n",
    "\\operatorname{det} A = a_{i1} C_{i1} + a_{i2} C_{i2} + \\cdots + a_{in} C_{in}\n",
    "$$\n",
    "$$\n",
    "\\operatorname{det} A = a_{1j} C_{1j} + a_{2j} C_{2j} + \\cdots + a_{nj} C_{nj}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font face=\"gotham\" color=\"purple\"> A SymPy Example of Determinant Expansion</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the matrix below and perform a cofactor expansion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}49 & 0 & 61\\\\73 & 22 & 96\\\\2 & 0 & 32\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡49  0   61⎤\n",
       "⎢          ⎥\n",
       "⎢73  22  96⎥\n",
       "⎢          ⎥\n",
       "⎣2   0   32⎦"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[49, 0, 61], [73, 22, 96], [2, 0, 32]]); A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cofactor expansion with the column which has two zero(s) involves the least computation burden:\n",
    "\n",
    "$$\\operatorname{det} A = a_{12}(-1)^{1+2}C_{12}+a_{22}(-1)^{2+2}C_{22}+a_{32}(-1)^{3+2}C_{32}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use SymPy function for calculationg minors: ```sy.matrices.matrices.MatrixDeterminant.minor(A, i, 1)```. Also we define a function for cofactor expansion:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cof_exp(matrix, c):\n",
    "    \"\"\"Calculate the cofactor expansion of a matrix along the c-th column.\"\"\"\n",
    "    detA = 0\n",
    "    for i in range(matrix.shape[0]):  # matrix.shape[0] is the total number of rows\n",
    "        minor_matrix = matrix.minor_submatrix(i, c)\n",
    "        cofactor = (-1)**(i + c) * minor_matrix.det()\n",
    "        detA += matrix[i, c] * cofactor\n",
    "    return detA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 31812$"
      ],
      "text/plain": [
       "31812"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cof_exp(A,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's easy to verify the expansion algorithm Sympy's determinant evaluation function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 31812$"
      ],
      "text/plain": [
       "31812"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Actually you can experiment with any random matrices with multiple zeros, the function below has the parameter ```percent=70``` which means $70\\%$ of element are non-zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 25 & 32 & 46 & 0 & 32 & 0\\\\15 & 0 & 0 & 44 & 41 & 0 & 0\\\\0 & 35 & 43 & 44 & 0 & 25 & 13\\\\18 & 30 & 25 & 0 & 32 & 36 & 17\\\\0 & 0 & 23 & 50 & 32 & 44 & 31\\\\0 & 36 & 29 & 11 & 0 & 42 & 0\\\\18 & 19 & 13 & 26 & 46 & 42 & 23\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0   25  32  46  0   32  0 ⎤\n",
       "⎢                          ⎥\n",
       "⎢15  0   0   44  41  0   0 ⎥\n",
       "⎢                          ⎥\n",
       "⎢0   35  43  44  0   25  13⎥\n",
       "⎢                          ⎥\n",
       "⎢18  30  25  0   32  36  17⎥\n",
       "⎢                          ⎥\n",
       "⎢0   0   23  50  32  44  31⎥\n",
       "⎢                          ⎥\n",
       "⎢0   36  29  11  0   42  0 ⎥\n",
       "⎢                          ⎥\n",
       "⎣18  19  13  26  46  42  23⎦"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = sy.randMatrix(r = 7, min=10, max=50, percent=70);B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate determinants with our user-defined function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle -7052024832$"
      ],
      "text/plain": [
       "-7052024832"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cof_exp(B, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then verify the result of using determinant method  ```.det()```. We can see indeed cofactor expansion works!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle -7052024832$"
      ],
      "text/plain": [
       "-7052024832"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Minor matrices_ can also be extracted by using code ```.minor_submatrix()```, for instance, the $M_{23}$ matrix of $  B$ is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 25 & 46 & 0 & 32 & 0\\\\0 & 35 & 44 & 0 & 25 & 13\\\\18 & 30 & 0 & 32 & 36 & 17\\\\0 & 0 & 50 & 32 & 44 & 31\\\\0 & 36 & 11 & 0 & 42 & 0\\\\18 & 19 & 26 & 46 & 42 & 23\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0   25  46  0   32  0 ⎤\n",
       "⎢                      ⎥\n",
       "⎢0   35  44  0   25  13⎥\n",
       "⎢                      ⎥\n",
       "⎢18  30  0   32  36  17⎥\n",
       "⎢                      ⎥\n",
       "⎢0   0   50  32  44  31⎥\n",
       "⎢                      ⎥\n",
       "⎢0   36  11  0   42  0 ⎥\n",
       "⎢                      ⎥\n",
       "⎣18  19  26  46  42  23⎦"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.minor_submatrix(1, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Cofactor matrix_ is the matrix contains all cofactors of original matrix, and function ```.cofactor_matrix()``` can easily produce this type of matrix.\n",
    "\n",
    "$$A=\\left[\\begin{array}{rrr}\n",
    "C_{11} & C_{12} & C_{13} \\\\\n",
    "C_{21} & C_{22} & C_{23} \\\\\n",
    "C_{31} & C_{32} & C_{33}\n",
    "\\end{array}\\right]=\n",
    "\\left[\\begin{array}{rrr}\n",
    "(-1)^{1+1}M_{11} & (-1)^{1+2}M_{12} & (-1)^{1+3}M_{13} \\\\\n",
    "(-1)^{2+1}M_{21} & (-1)^{2+2}M_{22} & (-1)^{2+3}M_{23} \\\\\n",
    "(-1)^{3+1}M_{31} & (-1)^{3+2}M_{32} & (-1)^{3+3}M_{33}\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}704 & -2144 & -44\\\\0 & 1446 & 0\\\\-1342 & -251 & 1078\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 704   -2144  -44 ⎤\n",
       "⎢                  ⎥\n",
       "⎢  0    1446    0  ⎥\n",
       "⎢                  ⎥\n",
       "⎣-1342  -251   1078⎦"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.cofactor_matrix()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Triangular Matrix</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $A$ is triangular matrix, cofactor expansion can be applied repetitively, the outcome will be a product of the elements on the principal diagonal. \n",
    "\n",
    "$$\n",
    "\\operatorname{det A}_{n\\times n} = \\prod_{i=1}^n a_{ii}\n",
    "$$\n",
    "\n",
    "where $a_{ii}$ is the diagonal element. \n",
    "\n",
    "Here is the proof, start with $A$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "A=\\left[\\begin{array}{cccc}\n",
    "a_{11} & a_{12} & \\cdots & a_{1 n} \\\\\n",
    "& a_{22} & \\cdots & a_{2 n} \\\\\n",
    "& & \\ddots & \\vdots\\\\\n",
    "& & & a_{n n}\\\\\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "Cofactor expanding on the first column, the sign of $a_{ii}$ is always positive because $(-1)^{2i}$\n",
    "$$\n",
    "a_{11} \\cdot \\left[\\begin{array}{cccc}\n",
    "a_{22} & a_{22} & \\cdots & a_{2 n} \\\\\n",
    "& a_{33} & \\cdots & a_{3 n} \\\\\n",
    "& & \\ddots & \\vdots\\\\\n",
    "& & &  a_{n n}\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "Continue the cofactor expansion\n",
    "$$\n",
    "\\operatorname{det} A=a_{11} a_{22} \\cdot \\left[\\begin{array}{cccc}\n",
    "a_{33} & a_{34} & \\cdots & a_{3n} \\\\\n",
    "& a_{44} & \\cdots & a_{4n} \\\\\n",
    "& & \\ddots & \\vdots \\\\\n",
    "& & & a_{nn}\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "Iterating the expansion, eventually\n",
    "$$\n",
    "\\text { det } A=a_{11} \\cdots a_{n-2, n-2} \\cdot \\left[\\begin{array}{cc}\n",
    "a_{n-1, n-1} & a_{n-1, n} \\\\\n",
    "& a_{n n}\n",
    "\\end{array}\\right]=a_{11} \\cdots a_{n n}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's verify with a numeric example, generate a random upper triangular matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[79., 66., 59., 58., 33.],\n",
       "       [ 0., 88., 17., 39., 72.],\n",
       "       [ 0.,  0., 27., 86., 57.],\n",
       "       [ 0.,  0.,  0., 47., 73.],\n",
       "       [ 0.,  0.,  0.,  0., 58.]])"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.round(np.random.rand(5,5)*100)\n",
    "A_triu = np.triu(A); A_triu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the determinant with ```np.linalg.det```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIcAAAAQCAYAAAA8hGzxAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjkuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/TGe4hAAAACXBIWXMAABJ0AAASdAHeZh94AAAFRElEQVR4nO3Za6xdVREH8F9NSTVI0ICkH1QeDShEofUBbYilgBYVgmBqTAwGSWglakC5gMTXdEx8YKRSUGODhKLyCR8NaMUqEiuINpEmYiAUC6XSKAJaUqgVpdcPa23cnLtPvfec3tMv/Scnc9aaNXtm7TVr1prZM8bHx+3HfnRhZruRmVtweJ+xj0fE7J7xS3AK5uIEHISbI+K8fgoHkWnJno6PYQFeiadwH1ZGxNqesWfiEhyHQ/AX/B4rIuKevWHXoHPJzFfj83hny7Y1yIj4x/+RPQ/frc2lEfHtPY0fRvdLOuSfRnb8vtox9jPKYs3FtknaN4iMzPwKfoG34FZcjZ/gVVjUM/Yq/Bhvwu1YiXvxHtxdX/A+mUtmzlGc9AJswNfwsOLI92TmIXuQfQ2+jmcmad9QumdOeALbI2L5JPV9Ao/hT8oOunM6ZDJzKS7HTVgWEc/18A9o/Z+Ny/A4jo+Iv7V4p+KXys753r6YC76Jw3BxRFzXsm1Ffd4XcFGvUGbOwI1KtPxhneNUMSXdXc4xaUTECy8jM6dFJjNnKUZv1eEY9Zn/bjUPVyLi79qO0ejOzB1KtBnKrkFk6s5djC34Ru/jsAwfzMyxiHi2h38xTlOi5GmTMnBI3V3OMauG3dfiWfwB6yPi+akatJfwDmUxr8Huepd4A3ZhQ8f94SE8hxMz89CIeLJhZOZC5V6wZgR2d+HUStdFxO42IyJ2ZObdygLOxx0NLzOPxZeVu9X6zJyycwyiu8s5ZvvfhafBI5l5QUT8agCjhsVbK92FjYpjvIDMXI8lEfEERMTfM/OTWIH7M3ONEorn4Gz8HB8ejekT8LpKN/XhP6Qs0DHqAmXmTGU9tuJTo9TdeyG9EacrDnIg3ohVOAI/zcwThjBuUBxW6eUYx9uU3X881mEhbmkLRMQ1eK/i/EtxJd6HP2N173EzQhxc6dN9+E3/K1p9n8M8fCgi/jlK3S+KHBHRe3D+ERdl5jMYw3KcO4SBg6Bx4P/g7IjYUtv3Zea5eBCnZOaC5ojJzCvwRVyr3O7/itfjS7g5M+dGxBUjnMNAyMyTlGhxdVf6Pd3oSmW78K1KF06XIXvA9ko3thwDRMRO/Kw2T4TMXISrcGtEXBoRD0fEzoi4V3HsbRjLzKOm3/QJaHbnwX34Tf/2epx8RzkGPjtK3U3HZLOVJyo9cOo2DY0HK93eh98Ubl5W6VmVTkgrI2JnZm5QnGSekuOPEs1cjunDP7rSTXh5a9yuPtnQ9Zl5vXJR/fhe1I3JR475lY76ZVIuR+M4LjO77G0uqI9UOqvSCelqT/+ElHgEaBx2ce9cMvMgnIyd+C3+hRv6/DZWsbtqezJHzlR0oxU5arq0tTe/zswjlHObiYWjaUdEPJqZtymZxiVKVa+xbTHOUKLK7bX710rVcllmroqIba3x71Jewi78ZiQTaCEiNmfmOiUr+Ciua7FTicyrWmtwYddzMnO5Evlu6iqf15rGAdjc1IAG0P2iY+X9ylm8Ho9ih5L+nYmXYq2eEnpmnoNzarP57rIgM1fX/09GxGXDytTJzMOKWufYiCPrc57HhRHRnKnfV8rsb8cDmfkj5UJ6rHLkzMCVEfHUPprLRxTHvLZ+K3oAJyl1iE34tOFxh1IMPFIpeg2kux1e7lS+R8zBB3CpUhK+C+fjrI7q5NzKO1/ZwXBUq29Jh+FTlomIx/BmJYIdrUSQRbgNJ0fED1pjd+PdSjn4fuV+MaYcjWtxRkSs3Bt2DTiXzcr3odXKwowp73wl5vc67d7EVHXP2P/Jfj/64b84flS2cwfdwAAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle 511681104.0$"
      ],
      "text/plain": [
       "511681104.0000005"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.det(A_triu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Extract the diagonal with ```np.diag()```, then calculate the product. We should expect the same results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 511681104.0$"
      ],
      "text/plain": [
       "511681104.0"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_diag = np.diag(A_triu)\n",
    "np.prod(A_diag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Properties of Determinants</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Determinants have a long list of properties, but they are mostly derived from cofactor expansion. There is no need to memorize them.\n",
    "\n",
    "1. Let $A$ be an $n \\times n$ square matrix. If one row of $A$ is multiplied by $k$ to produce the matrix $B$, then $\\text{det} B = k \\, \\text{det} A$.\n",
    "2. Let $A$ be an $n \\times n$ square matrix. If two rows of $A$ are interchanged to produce a matrix $B$, then $\\text{det} B = -\\text{det} A$.\n",
    "3. Let $A$ be an $n \\times n$ square matrix. If a multiple of one row of $A$ is added to another row to produce the matrix $B$, then $\\text{det} A = \\text{det} B$.\n",
    "4. If $A$ is an $n \\times n$ matrix, then $\\text{det} A^T = \\text{det} A$.\n",
    "5. A square matrix $A$ is invertible if and only if $\\text{det} A \\neq 0$.\n",
    "6. If $A$ and $B$ are $n \\times n$ matrices, then $\\text{det} AB = (\\text{det} A)(\\text{det} B)$.\n",
    "7. If $A$ is an $n \\times n$ matrix and $k$ is a scalar, then $\\text{det} \\, kA = k^n \\, \\text{det} A$.\n",
    "8. If $A$ is an invertible square matrix, then $\\text{det} A^{-1} = \\frac{1}{\\text{det} A}$.\n",
    "\n",
    "All of these properties are straightforward. The key is to demonstrate them using cofactor expansion. Here are some casual proofs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font face=\"gotham\" color=\"red\">Proof of property 6</font>:\n",
    "$$\\begin{aligned}\n",
    "|A B| &=\\left|E_{p} \\cdots E_{1} B\\right|=\\left|E_{p}\\right|\\left|E_{p-1} \\cdots E_{1} B\\right|=\\cdots \\\\\n",
    "&=\\left|E_{p}\\right| \\cdots\\left|E_{1}\\right||B|=\\cdots=\\left|E_{p} \\cdots E_{1}\\right||B| \\\\\n",
    "&=|A||B|\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font face=\"gotham\" color=\"red\">Proof of property 7</font>:\n",
    "\n",
    "Because $\\text{det} B = k\\, \\text{det} A$, one row of $A$ is multiplied by $k$ to produce $B$.Then multiply all the rows of $A$ by $k$, there will be $n$ $k$'s in front of $\\text{det} A$, which is $k^n \\text{det} A$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font face=\"gotham\" color=\"red\">Proof of property 8</font>:\n",
    "$$\\begin{aligned}\n",
    "&\\begin{aligned}\n",
    "A A^{-1} &=I \\\\\n",
    "\\left|A A^{-1}\\right| &=|I|\n",
    "\\end{aligned}\\\\\n",
    "&|A|\\left|A^{-1}\\right|=1\\\\\n",
    "&\\left|A^{-1}\\right|=\\frac{1}{|A|}\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These properties are useful in the analytical derivation of other theorems; however, they are not efficient for numerical computation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Cramer's Rule</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If a linear system has $n$ equations and $n$ variables, an algorithm called _Cramer's Rule_ can solve the system in terms of determinants as long as the solution is unique.\n",
    "$$\n",
    "A_{n\\times n}\\mathbf{b}_{n} = \\mathbf{x}_n\n",
    "$$\n",
    "\n",
    "Some convenient notations are introduced here:\n",
    "\n",
    "For any $A_{n\\times n}$ and vector $\\mathbf{b}$, denote $A_i(\\mathbf{b})$ as the matrix obtained from replacing the $i$th column of $A$ by $\\mathbf{b}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$A_{i}(\\mathbf{b})=\\left[\\begin{array}{lllll}\n",
    "\\mathbf{a}_{1} & \\cdots & \\mathbf{b} & \\cdots & \\mathbf{a}_{n}\n",
    "\\end{array}\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Cramer's Rule can solve each $x_i$ without solving the whole system\n",
    "$$x_{i}=\\frac{\\operatorname{det} A_{i}(\\mathbf{b})}{\\operatorname{det} A}, \\quad i=1,2, \\ldots, n$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"background-color:Bisque; color:DarkBlue; padding:30px;\">\n",
    "<b>Fast Proof of Cramer's Rule</b>：\n",
    "\n",
    "$$\\begin{aligned}\n",
    "A \\cdot I_{i}(\\mathbf{x}) &=A\\left[\\mathbf{e}_{1} \\quad \\cdots \\quad \\mathbf{x} \\quad \\cdots \\quad \\mathbf{e}_{n}\\right]=\\left[\\begin{array}{llllll}\n",
    "A \\mathbf{e}_{1} & \\cdots & A \\mathbf{x} & \\cdots & A \\mathbf{e}_{n}\n",
    "\\end{array}\\right] \\\\\n",
    "&=\\left[\\begin{array}{llllll}\n",
    "\\mathbf{a}_{1} & \\cdots & \\mathbf{b} & \\cdots & \\mathbf{a}_{n}\n",
    "\\end{array}\\right]=A_{i}(\\mathbf{b})\n",
    "\\end{aligned}$$\n",
    "    \n",
    "where $I_i(\\mathbf{x})$ is an identity matrix whose $i$-th column replaced by $\\mathbf{x}$. With determinant's property,\n",
    "$$(\\operatorname{det} A)\\left(\\operatorname{det} I_{i}(\\mathbf{x})\\right)=\\operatorname{det} A_{i}(\\mathbf{b})$$\n",
    "\n",
    "$\\text{det}I_{i}(\\mathbf{x})=x_i$, can be shown by cofactor expansion.\n",
    "</div> "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> A NumPy Example On Cramer's Rule</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the system\n",
    "$$\\begin{aligned}\n",
    "&2 x-y+3 z=-3\\\\\n",
    "&3 x+3 y-z=10\\\\\n",
    "&-x-y+z=-4\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You have surely known several ways to solve it, but let's test if Cramer's rule works. \n",
    "\n",
    "Input the matrices into NumPy arrays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[2, -1, 3], [3, 3, -1], [-1, -1, 1]])\n",
    "b = np.array([-3, 10, -4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [],
   "source": [
    "A_1b = np.copy(A) # Python variable is a reference tag\n",
    "A_1b[:,0]=b\n",
    "\n",
    "A_2b = np.copy(A)\n",
    "A_2b[:,1]=b\n",
    "\n",
    "A_3b = np.copy(A)\n",
    "A_3b[:,2]=b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to Cramer's rule:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAUCAYAAAAdtqaHAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjkuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/TGe4hAAAACXBIWXMAABJ0AAASdAHeZh94AAAEh0lEQVR4nO2aa4hWRRjHf6sGmcgKRglGtW1qQdLShYzQWta0CMLK6EteCqIo2wylsAsP/8DyQ61ldIEWfCOCCpaKUkooobKLJQlFmmYt3QxTy8yKats+zJz17Nnztmffs+ey9v7gZd7znDMzzzPzzJmZZ05Db28vdeqUnVFFK1CnThLGxAklPQ1cCjSZ2aF8Varzf0XS2cBHwA1m1hm+1xCd+iWdC3wALDezjpB8PnAh0AKcCYwHnjWza2tU6gTgPuASYCKwG3gJkJn9VEuZg9Q3EbgCuAyYDkwG/gQ+AdYCa83snyGWmasNeVB0P0t6EZgBTDGzXwN53NS/EvgFeCIivwdY4g34rhalQ8o0A1uA64DNwGrgS+A24D3vVMPN1cBTwHm4gfgw0AWcAXQCL0hqSFpYQTbkQdH9/AAwCWgPC/tN/ZKmArOBTjP7PVLA7cC3wBe4EbcxhQ2PA8cB7Wb2aKj+Dl/PSuCmFOXHsQO4HFgXfnNKugvXiFcBV+KcNwlF2JAHhfazmW2WtB24UdKqoK+ib9TrgQbg+WiNZrbRzHaaWaowgR9lc4Bu4LFoNcAhYIGkcWnqiWJmb5rZK9Hp3cx+AJ70lxclKasoG/KgJP38HHAicHEgiDrqbKAHeD+NkoPQ6tMNMU5zENgEHINbp+TFXz79O+HzZbShbKRpo00+Heio3rNbgG0Z7/Sn+XRHlfs7fTo1Qx36kDQGWOgvX0uYrVQ2lJQ0bfShT2cFgvAbdTIwGrcry5JGnx6ocj+QT8hYj4BVuA3VejN7PWGestlQRmpuIzM7APyBm/6B/pupYAc2IsMqtSCpHVgGbAcWFKzOkJHUDZw0hCw1h5kKYD9wfHARdtRgl390xgoEI6mxyv1A/nOWSkhaAjwCfAa0mdn+IWQvhQ3ALtybJynfZ6VIDGnbaCyHfbKfo+7xadbxv899Wm39NsWn1dY2qZG0FBfT+xTnpHv+O8cACrcBwMzasiw/JTW3kaRRuCXBV4EsvEbdDfzI4UVwVgRxuTleobCC44ELgN/IKPIg6U6ck24FWmtwUijYhhFCmjaahguTbg0EfQX4uNlbwLGSTh0OTSU1SzpN0lGhenYBG4CTgVuiWYBxwDPRyIOkiqReSYtT6HMvbvO0Bfcm3TvSbCgjw9lGniBk1XfYEP0opQt3QjMXdzIRVmYeMM9fTvLp+ZIq/v9eM1seKe8N3GK/CRf4DbgZeBdYI6kN2IY72mzFTQV3xygfDKqksc5+SFqEO3PuAd4G2iVFH+s2s0pZbciDEvQzuIOCHuDlQBAN+Hfh1qoLGUgLsMj/5nrZKSHZ/CqVDsCPtnOAild8GdCM29zMMLN9MdmmAweBdUnridDk09HAUtzpSPS3OGlhBdmQBy0U2M+SGnED5VUz+yaQx309tQK4HzjLzD5OqlSWSJoA7AMeMrM7ClanJo4EG/JA0q3AGmCmmb0TyOO+nloNfI2bJsvCTNwxZ8dgD5aYI8GGTJE0FlgBdIWdFGLeqD7DLNw64sH6h9N18kLS6cA1QMXMusP3/gXV+w5nabyUxQAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left( 1.0, \\  2.0, \\  -1.0\\right)$"
      ],
      "text/plain": [
       "(1.0000000000000004, 1.9999999999999982, -1.0000000000000002)"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x1 = np.linalg.det(A_1b)/np.linalg.det(A)\n",
    "x2 = np.linalg.det(A_2b)/np.linalg.det(A)\n",
    "x3 = np.linalg.det(A_3b)/np.linalg.det(A)\n",
    "(x1, x2, x3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can verify the results by NumPy built-in function ```np.linalg.solve```."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.,  2., -1.])"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.solve(A, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or in a straightforward way $A^{-1}b$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.,  2., -1.])"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.inv(A)@b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All results are the same!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, remember that Cramer's rule is rarely used in practice for solving systems of equations, as its computational cost (measured by the number of floating-point operations, or flops) is much higher than that of Gaussian-Jordan elimination."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> A Determinant Formula For Inverse Matrix</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An alternative algorithm for $A^{-1}$ is\n",
    "$$A^{-1}=\\frac{1}{\\operatorname{det} A}\\left[\\begin{array}{cccc}\n",
    "C_{11} & C_{21} & \\cdots & C_{n 1} \\\\\n",
    "C_{12} & C_{22} & \\cdots & C_{n 2} \\\\\n",
    "\\vdots & \\vdots & & \\vdots \\\\\n",
    "C_{1 n} & C_{2 n} & \\cdots & C_{n n}\n",
    "\\end{array}\\right]$$\n",
    "\n",
    "where the matrix of cofactors on RHS is the _adjugate_ matrix, SymPy function is ```sy.matrices.matrices.MatrixDeterminant.adjugate```. And this is the transpose of the _cofactor matrix_ which we computed using ```sy.matrices.matrices.MatrixDeterminant.cofactor_matrix```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> A SymPy Example</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate a random matrix with $20\\%$ of zero elements."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 2 & -1 & 4 & -5\\\\-1 & 0 & 0 & -5 & 4\\\\0 & 5 & 0 & 1 & -5\\\\-2 & -5 & 5 & 2 & 1\\\\0 & -3 & 0 & 0 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0   2   -1  4   -5⎤\n",
       "⎢                  ⎥\n",
       "⎢-1  0   0   -5  4 ⎥\n",
       "⎢                  ⎥\n",
       "⎢0   5   0   1   -5⎥\n",
       "⎢                  ⎥\n",
       "⎢-2  -5  5   2   1 ⎥\n",
       "⎢                  ⎥\n",
       "⎣0   -3  0   0   4 ⎦"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.randMatrix(5, min=-5, max = 5, percent = 80); A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the adjugate matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-185 & -162 & 4 & -37 & -55\\\\-20 & 8 & 128 & -4 & 128\\\\-101 & -54 & 80 & 27 & 21\\\\25 & -10 & 76 & 5 & 135\\\\-15 & 6 & 96 & -3 & 155\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-185  -162   4   -37  -55⎤\n",
       "⎢                         ⎥\n",
       "⎢-20    8    128  -4   128⎥\n",
       "⎢                         ⎥\n",
       "⎢-101  -54   80   27   21 ⎥\n",
       "⎢                         ⎥\n",
       "⎢ 25   -10   76    5   135⎥\n",
       "⎢                         ⎥\n",
       "⎣-15    6    96   -3   155⎦"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_adjugate = A.adjugate(); A_adjugate "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can verify if this really the adjugate of $A$, we pick element of $(1, 3), (2, 4), (5, 4)$ of $A$ to compute the cofactors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'(2, 0) minor: 4'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'(3, 1) minor: -4'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'(3, 4) minor: -3'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(f\"(2, 0) minor: {(-1)**(1+3) * A.minor(2, 0)}\") # (1, 3) cofactor of A, transposed index\n",
    "display(f\"(3, 1) minor: {(-1)**(2+4) * A.minor(3, 1)}\")\n",
    "display(f\"(3, 4) minor: {(-1)**(5+4) * A.minor(3, 4)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adjugate is the transpose of cofactor matrix, thus we reverse the row and column index when referring to the elements. As we have shown it is truel adjugate matrix. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we will check if this inverse formula works with SymPy's function. \n",
    "\n",
    "The ```sy.N()``` is for converting to float approximation, i.e. if you don't like fractions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-0.7839 & -0.6864 & 0.0169 & -0.1568 & -0.2331\\\\-0.0847 & 0.0339 & 0.5424 & -0.0169 & 0.5424\\\\-0.428 & -0.2288 & 0.339 & 0.1144 & 0.089\\\\0.1059 & -0.0424 & 0.322 & 0.0212 & 0.572\\\\-0.0636 & 0.0254 & 0.4068 & -0.0127 & 0.6568\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-0.7839  -0.6864  0.0169  -0.1568  -0.2331⎤\n",
       "⎢                                          ⎥\n",
       "⎢-0.0847  0.0339   0.5424  -0.0169  0.5424 ⎥\n",
       "⎢                                          ⎥\n",
       "⎢-0.428   -0.2288  0.339   0.1144    0.089 ⎥\n",
       "⎢                                          ⎥\n",
       "⎢0.1059   -0.0424  0.322   0.0212    0.572 ⎥\n",
       "⎢                                          ⎥\n",
       "⎣-0.0636  0.0254   0.4068  -0.0127  0.6568 ⎦"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_det = A.det()\n",
    "A_inv = (1/A_det)*A_adjugate\n",
    "round_expr(sy.N(A_inv), 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now again, we can verify the results with ```.inv()```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-0.7839 & -0.6864 & 0.0169 & -0.1568 & -0.2331\\\\-0.0847 & 0.0339 & 0.5424 & -0.0169 & 0.5424\\\\-0.428 & -0.2288 & 0.339 & 0.1144 & 0.089\\\\0.1059 & -0.0424 & 0.322 & 0.0212 & 0.572\\\\-0.0636 & 0.0254 & 0.4068 & -0.0127 & 0.6568\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-0.7839  -0.6864  0.0169  -0.1568  -0.2331⎤\n",
       "⎢                                          ⎥\n",
       "⎢-0.0847  0.0339   0.5424  -0.0169  0.5424 ⎥\n",
       "⎢                                          ⎥\n",
       "⎢-0.428   -0.2288  0.339   0.1144    0.089 ⎥\n",
       "⎢                                          ⎥\n",
       "⎢0.1059   -0.0424  0.322   0.0212    0.572 ⎥\n",
       "⎢                                          ⎥\n",
       "⎣-0.0636  0.0254   0.4068  -0.0127  0.6568 ⎦"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round_expr(sy.N(A.inv()), 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or We can show by difference."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0  0  0  0  0⎤\n",
       "⎢             ⎥\n",
       "⎢0  0  0  0  0⎥\n",
       "⎢             ⎥\n",
       "⎢0  0  0  0  0⎥\n",
       "⎢             ⎥\n",
       "⎢0  0  0  0  0⎥\n",
       "⎢             ⎥\n",
       "⎣0  0  0  0  0⎦"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_inv-A.inv()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So Cramer's rule indeed works perfectly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"background-color:Bisque; color:DarkBlue; padding:30px;\">\n",
    "\n",
    "<b>Short Proof of $A^{-1}$ Formula With Determinants</b><br>\n",
    "\n",
    "We define $x$ the $j$-th column of $A^{-1}$ which satisfies\n",
    "$$\n",
    "Ax= e_j\n",
    "$$\n",
    "\n",
    "and $e_j$ is the $j$-th column of an identity matrix, and $j$th entry of $x$ is the $(i,j)$-entry of $A^{-1}$. By Cramer's rule,\n",
    "\n",
    "$$\\left\\{(i, j) \\text { -entry of } A^{-1}\\right\\}=x_{i}=\\frac{\\operatorname{det} A_{i}\\left(\\mathbf{e}_{j}\\right)}{\\operatorname{det} A}$$\n",
    "\n",
    "The cofactor expansion along column $i$ of $A_i(e_j)$,\n",
    "$$\\operatorname{det} A_{i}\\left(\\mathbf{e}_{j}\\right)=(-1)^{i+j} \\operatorname{det} A_{j i}=C_{j i}$$\n",
    "</div>"
   ]
  }
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