linearalgebra_practice101102103.ipynb

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   "execution_count": 3,
   "metadata": {},
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  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math, sympy, numpy\n",
    "sympy.init_printing(use_unicode=True)\n",
    "from sympy import *\n",
    "import pylab as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет 1 / Standard Calculation 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "\n",
    "По данным матрицам $A,B,C,D$ и числам $\\alpha$ и $\\beta$ вычислить матрицу $ G = \\alpha * A * B \\ + \\ \\beta * C^T * D$.\n",
    "\n",
    "$\\alpha = 6, \\ \\beta = -7, \\\\ \n",
    "A_{(2,4)} = \n",
    " \\begin{pmatrix}\n",
    "-3 & -3 & -6 & -3\\\\\n",
    "-4 & -7 & 8 & -7\\\\\n",
    " \\end{pmatrix}, \\ \n",
    "B_{(4,3)} = \n",
    " \\begin{pmatrix}\n",
    "2 & 10 & 7 \\\\\n",
    "-5 & -6 & -2 \\\\\n",
    "-8 & 4 & -8 \\\\\n",
    "7 & 4 & -6 \\\\\n",
    " \\end{pmatrix}, \\ \n",
    "C_{(2,2)} = \n",
    " \\begin{pmatrix}\n",
    "-8 & -4 \\\\\n",
    "3 & -5 \\\\\n",
    " \\end{pmatrix}, \\ \n",
    "D_{(2,3)} = \n",
    " \\begin{pmatrix}\n",
    "-2 & -1 & -5 \\\\\n",
    "-2 & -10 & 8 \\\\\n",
    " \\end{pmatrix}.\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "α, β = 6, -7\n",
    "A = sympy.Matrix([[-3, -3, -6,-3], [-4, -7, 8,-7]])\n",
    "B = sympy.Matrix([[2, 10, 7], [-5, -6, -2], [-8, 4, -8], [7, 4, -6]])\n",
    "C = sympy.Matrix([[-8, -4], [3, -5]])\n",
    "D = sympy.Matrix([[-2, -1, -5], [-2, -10, 8]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decisions:\n",
    "1) Рассчитаем элементы матрицы $\\alpha * A * B$:\n",
    "\n",
    "$\\alpha * A * B|_{11} = 6 * (-3 * 2 - 3 * (-5)  - 6 * (-8) - 3 * 7) = 216 \\ \\| \\ \\alpha \\ * $ сумму элементов 1-ой строки $A \\ * $ на элементы 1-ого столбца $B$.\n",
    "\n",
    "$\\alpha * A * B|_{12} = 6 * (-3 * 10 - 3 * (-6)  - 6 * 4 - 3 * 4) = -288$\n",
    "\n",
    "$\\alpha * A * B|_{13} = 6 * (-3 * 7 - 3 * (-2)  - 6 * (-8) - 3 * (-6)) = 306$\n",
    "\n",
    "$\\alpha * A * B|_{21} = 6 * (-4 * 2 - 7 * (-5)  + 8 * (-8) - 7 * 7) = -516$\n",
    "\n",
    "$\\alpha * A * B|_{22} = 6 * (-4 * 10 - 7 * (-6)  + 8 * 4 - 7 * 4) = 36$\n",
    "\n",
    "$\\alpha * A * B|_{23} = 6 * (-4 * 7 - 7 * (-2)  + 8 * (-8) - 7 * (-6)) = -216$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left ( 216, \\quad -288, \\quad 306\\right )$$"
      ],
      "text/plain": [
       "(216, -288, 306)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "6 * (-3 * 2 - 3 * (-5)  - 6 * (-8) - 3 * 7), 6 * (-3 * 10 - 3 * (-6)  - 6 * 4 - 3 * 4), 6 * (-3 * 7 - 3 * (-2)  - 6 * (-8) - 3 * (-6)) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left ( -516, \\quad 36, \\quad -216\\right )$$"
      ],
      "text/plain": [
       "(-516, 36, -216)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "6 * (-4 * 2 - 7 * (-5)  + 8 * (-8) - 7 * 7), 6 * (-4 * 10 - 7 * (-6)  + 8 * 4 - 7 * 4), 6 * (-4 * 7 - 7 * (-2)  + 8 * (-8) - 7 * (-6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2) Запишем транспонированную матрицу $C^T$:\n",
    "    \n",
    "$C_{(2,2)}^T = \n",
    " \\begin{pmatrix}\n",
    "-8 & 3 \\\\\n",
    "-4 & -5 \\\\\n",
    " \\end{pmatrix}$\n",
    " \n",
    " Элементы 1-ой строки $C$ стали элементами 1-ого столбца $C^T$, \n",
    " \n",
    " элементы 2-ой строки $C$ - элементами 2-ого столбца $C^T$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-8 & 3\\\\-4 & -5\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-8  3 ⎤\n",
       "⎢      ⎥\n",
       "⎣-4  -5⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3) Рассчитаем элементы матрицы $\\beta * C^T * D$:\n",
    "    \n",
    "$\\beta * C^T * D |_{11} = -7 * (-8 * (-2) + 3 * (-2)) = -70$\n",
    "\n",
    "$\\beta * C^T * D |_{12} = -7 * (-8 * (-1) + 3 * (-10)) = 154$\n",
    "\n",
    "$\\beta * C^T * D |_{13} = -7 * (-8 * (-5) + 3 * 8) = -448$\n",
    "\n",
    "$\\beta * C^T * D |_{21} = -7 * (-4 * (-2) - 5 * (-2)) = -126$\n",
    "\n",
    "$\\beta * C^T * D |_{22} = -7 * (-4 * (-1) - 5 * (-10)) = -378$\n",
    "\n",
    "$\\beta * C^T * D |_{23} = -7 * (-4 * (-5) - 5 * 8) = 140$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left ( -70, \\quad 154, \\quad -448\\right )$$"
      ],
      "text/plain": [
       "(-70, 154, -448)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-7 * (-8 * (-2) + 3 * (-2)), -7 * (-8 * (-1) + 3 * (-10)), -7 * (-8 * (-5) + 3 * 8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( -126, \\quad -378, \\quad 140\\right )$$"
      ],
      "text/plain": [
       "(-126, -378, 140)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-7 * (-4 * (-2) - 5 * (-2)), -7 * (-4 * (-1) - 5 * (-10)), -7 * (-4 * (-5) - 5 * 8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}216 & -288 & 306\\\\-516 & 36 & -216\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡216   -288  306 ⎤\n",
       "⎢                ⎥\n",
       "⎣-516   36   -216⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "α * A * B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-70 & 154 & -448\\\\-126 & -378 & 140\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-70   154   -448⎤\n",
       "⎢                ⎥\n",
       "⎣-126  -378  140 ⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "β * C.transpose() * D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4) Сложим полученные матрицы $α * A * B + β * C^T * D$:\n",
    "    \n",
    "$G_{11} = 216 - 70 = 146; G_{12} = -288 + 154 = -134; G_{13} = 306 - 448 = -142$\n",
    "\n",
    "$G_{21} = -516 - 126 = -642; G_{22} = 36 - 378 = -342; G_{23} = -216 + 140 = -76$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}146 & -134 & -142\\\\-642 & -342 & -76\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡146   -134  -142⎤\n",
       "⎢                ⎥\n",
       "⎣-642  -342  -76 ⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = α * A * B + β * C.transpose() * D\n",
    "G"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет 2 / Standard Calculation 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "\n",
    "Для данной матрицы А найти обратную $А^{-1}$ и выполнить проверку результата.\n",
    "\n",
    "For a given matrix A, find the inverse $A^{- 1}$ and perform a result check.\n",
    "\n",
    "$\n",
    "A = \n",
    " \\begin{pmatrix}\n",
    "20 & 19 & 9 \\\\\n",
    "31 & 30 & 22 \\\\\n",
    "26 & 25 & 16 \\\\\n",
    " \\end{pmatrix}\n",
    " $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = sympy.Matrix([[20,19,9],[31,30,22],[26,25,16]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decisions:\n",
    "\n",
    "1) Вычислим определитель матрицы A.\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "20 & 19 & 9 \\\\\n",
    "31 & 30 & 22 \\\\\n",
    "26 & 25 & 16 \\\\\n",
    "\\end{pmatrix}$\n",
    " \n",
    " Из 2-ой и 3-ей строки вычтем 1-ую:\n",
    " \n",
    "$\\begin{pmatrix}\n",
    "20 & 19 & 9 \\\\\n",
    "11 & 11 & 13 \\\\\n",
    "6 & 6 & 7 \\\\\n",
    "\\end{pmatrix}$\n",
    " \n",
    " Из 2-ого и 3-его столбца вычтем 1-ый:\n",
    " \n",
    "$\\begin{pmatrix}\n",
    "20 & -1 & -11 \\\\\n",
    "11 & 0 & 2 \\\\\n",
    "6 & 0 & 1 \\\\\n",
    "\\end{pmatrix}$\n",
    " \n",
    "Разложим по 2-му столбцу:\n",
    "\n",
    "$ \\Delta_A = -(-1) * \\begin{vmatrix}\n",
    "11 & 2 \\\\\n",
    "6 & 1 \\\\\n",
    "\\end{vmatrix} = 11 * 1 - 6 * 2 = -1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}20 & 19 & 9\\\\31 & 30 & 22\\\\26 & 25 & 16\\end{matrix}\\right], \\quad -1\\right )$$"
      ],
      "text/plain": [
       "⎛⎡20  19  9 ⎤    ⎞\n",
       "⎜⎢          ⎥    ⎟\n",
       "⎜⎢31  30  22⎥, -1⎟\n",
       "⎜⎢          ⎥    ⎟\n",
       "⎝⎣26  25  16⎦    ⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A, A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}20 & 19 & 9\\\\11 & 11 & 13\\\\6 & 6 & 7\\end{matrix}\\right], \\quad -1\\right )$$"
      ],
      "text/plain": [
       "⎛⎡20  19  9 ⎤    ⎞\n",
       "⎜⎢          ⎥    ⎟\n",
       "⎜⎢11  11  13⎥, -1⎟\n",
       "⎜⎢          ⎥    ⎟\n",
       "⎝⎣6   6   7 ⎦    ⎠"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sympy.Matrix([[20,19,9],[31-20,30-19,22-9],[26-20,25-19,16-9]])\n",
    "A, A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}20 & -1 & -11\\\\11 & 0 & 2\\\\6 & 0 & 1\\end{matrix}\\right], \\quad -1\\right )$$"
      ],
      "text/plain": [
       "⎛⎡20  -1  -11⎤    ⎞\n",
       "⎜⎢           ⎥    ⎟\n",
       "⎜⎢11  0    2 ⎥, -1⎟\n",
       "⎜⎢           ⎥    ⎟\n",
       "⎝⎣6   0    1 ⎦    ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sympy.Matrix([[20,19-20,9-20],[11,11-11,13-11],[6,6-6,7-6]])\n",
    "A, A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}11 & 2\\\\6 & 1\\end{matrix}\\right], \\quad -1\\right )$$"
      ],
      "text/plain": [
       "⎛⎡11  2⎤    ⎞\n",
       "⎜⎢     ⎥, -1⎟\n",
       "⎝⎣6   1⎦    ⎠"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A1 = sympy.Matrix([[11,2],[6,1]])\n",
    "A1, - (-1) * A1.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2) Вычислим матрицу алгебраических дополнений:\n",
    "    \n",
    "$A_S = \n",
    "\\begin{pmatrix}\n",
    "30*16-25*22 & -(31*16-26*22) & 31*25-26*30 \\\\\n",
    "-(19*16-25*9) & 20*16-26*9 & -(20*25-26*19) \\\\\n",
    "19*22-30*9 & -(20*22-31*9) & 20*30-31*19\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "-70 & 76 & -5 \\\\\n",
    "-79 & 86 & -6 \\\\\n",
    "148 & -161 & 11\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( -70, \\quad 76, \\quad -5\\right )$$"
      ],
      "text/plain": [
       "(-70, 76, -5)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "30*16-25*22, -(31*16-26*22), 31*25-26*30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( -79, \\quad 86, \\quad -6\\right )$$"
      ],
      "text/plain": [
       "(-79, 86, -6)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-(19*16-25*9), 20*16-26*9, -(20*25-26*19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 148, \\quad -161, \\quad 11\\right )$$"
      ],
      "text/plain": [
       "(148, -161, 11)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "19*22-30*9, -(20*22-31*9), 20*30-31*19"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAABLCAMAAADTRTyqAAAAOVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACXHtMAAAAEnRSTlMAMquJZkQiEEAwzd1U75m7dnyNtFPXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFAElEQVRoBe1bi7KcIAxFcaX1Xf7/Y5sEMMDyuqvT3k5lppWNOckxIsLZvaLT1HrxTdts+AnRaTlAe31TngLJDb1GomOa4zio9Imy9UNYHLRDg3LUhhLRxdRcr4CQ8tgcKA4ZffZhgJOSUkZO9Y+rnqZ5dqUqEpVTv0GbFyEm+KfmNqYebJxhQPVznVXCY571LB1PUSS6EXyEw0C55J6I925imJiwv7fB4kDhw10kClWEhoCdUIs+L5DOZP5jWCMgHecLRCkAjTBNVXppwyEdOLQSbMLR/Wn7ItER75tyRIfWrAQTeurgGfxw3uuH49jO57B465FVj2lGLbHf6QMPLY1gSk94ZWv7ffBD7zDORu2uskZ0hOnLI0p8/WiZvoEpTehtbRrayVC7mzECogrmrbOZossJ8cpU9NVcUQMTmtIMZ1mSXKwxkRzObO5lFBBNxJnN1GLG6NL8MFnYTJc56Oah7THYCbtpO0orRJU2z56ZCoe26QnvgIV9oaIeR+quhMVXPLUK0ZcppRhontnoIi2ydHCwlx2jJd/cOfM4rO5lUSG6mMEpBL5G1XpOFrno1n7CdkinPnvq8YUojvM5rBAdXRIlj6Nv5SlOmJDb1g4LL38E6HbOFxWiIfRvfnqI3l39p6JPRe+uwN3xnjH6VDRVgZuUhFToL9uKY9RXEmB/358v3koaWBhIt1P6XIBA0YPFiyJRVhLUDLulbm0TIARtWXvcKV0RIEJskSglFLje6mk9uhtDpZ4gaqGHwsXrFQEixBaJmt0jLtY1LZmlW24jkXwzggoSvSJARNgiUeKCSoLStNA+3AYmTxLPvDTudDeo6xUBIsJWiTolARnIpu0kDBStjw1vxxUBIsJWiZKS4MZo63ZSaj3Bg3dFgIixNaJGSYB9DygWXd+4XV7kuGuo/hUBIsYGRBMagFUShNoOuTSO0QEnh0XDvuyKABFhkegP/RNHYLJZJcGc29r29WbnOEL5rwgQEfYXEM1q+DjKfPFvcpvs5DU5o7KTWD8Ioxy1SToO7o4RNrj1zoePTklYsEpj2+wk7AtsH2GiwlDbRypphK0QdUrCgaL/3ighvSZcFAwoUV4QICJsheipJMDywC0zsEjl1uF3FEbYvCBAhOJFhWiZ0J88+xC9u9pPRZ+K3l2Bu+M9Y/Sp6N0VuDve+xjtzh8dwAKQ9se+ntCcn8OwjMC2bBh24R45R0RVv03e6nim1SixJT0hGz864YU5ZQTPFnmfH9mFe+5kRBTMBxM9iKinJzhUw9GF8WUEZyvA2YV75F4i2g0TVpT1hEKCt1M2TyAjRLnfMGBgF+6RX4moFESU9YRU5JzN5glkhCh3Csou3KsRPUZDlPWEVOSczeYJZIQodwrKLtyrEB1RkjFbO6snpAJnbSZPKCNEuVNYduFehSh+u2uIOj0hFThrc0Tt9g53Ud4ArMASztkxOqAWSkRZT8iGTygXtiCBjBAVKRWOXbhHfjmiI/3MhYiynpCKnLPZPIGMEOVOQdmFe0Wirx6bXnvp6QmpyDmbzRPICFHuFJRduFckaoKs+DCxnpCKnLPZPIGMEOVOQdmFe+SXu/UmCOm3np6QCp2xuTy+BOFsGQia2YV75B4T7fdVz735HRa89vWOXzKwnlBIEZzywpwShGcLfL0P7MI9ezom6qG+V/chevf9eCr6n1f0H/mDAUW/yP/mfzAAG6LfMQE9o3MbUb8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-70 & 76 & -5\\\\-79 & 86 & -6\\\\148 & -161 & 11\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-70   76   -5⎤\n",
       "⎢             ⎥\n",
       "⎢-79   86   -6⎥\n",
       "⎢             ⎥\n",
       "⎣148  -161  11⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Матрица алгебраических дополнений / Matrix of algebraic complements\n",
    "A = sympy.Matrix([[20,19,9],[31,30,22],[26,25,16]])\n",
    "A.adjugate().T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "def minor(m,i,j):\n",
    "    return [row[:j] + row[j+1:] for row in (m[:i] + m[i+1:])]\n",
    "\n",
    "def deternminant(m):\n",
    "    if len(m) == 2:\n",
    "        return m[0][0] * m[1][1] - m[0][1] * m[1][0]\n",
    "\n",
    "    det = 0\n",
    "    for c in range(len(m)):\n",
    "        det += ((-1) ** c) * m[0][c] * deternminant(minor(m,0,c))\n",
    "    return det"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 1  -  [[30, 22], [25, 16]]  :  -70\n",
      "1 2  -  [[31, 22], [26, 16]]  :  -76\n",
      "1 3  -  [[31, 30], [26, 25]]  :  -5\n",
      "2 1  -  [[19, 9], [25, 16]]  :  79\n",
      "2 2  -  [[20, 9], [26, 16]]  :  86\n",
      "2 3  -  [[20, 19], [26, 25]]  :  6\n",
      "3 1  -  [[19, 9], [30, 22]]  :  148\n",
      "3 2  -  [[20, 9], [31, 22]]  :  161\n",
      "3 3  -  [[20, 19], [31, 30]]  :  11\n"
     ]
    }
   ],
   "source": [
    "AA = [[20,19,9],[31,30,22],[26,25,16]]\n",
    "for i in range(3):\n",
    "    for j in range(3):\n",
    "        print(i+1, j+1, ' - ', minor(AA,i,j), ' : ', deternminant(minor(AA,i,j)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3) Запишем транспонированную матрицу алгебраических дополнений\n",
    "\n",
    "$A_S^T = \n",
    "\\begin{pmatrix}\n",
    "-70 & -79 & 148 \\\\\n",
    "76 & 86 & -161 \\\\\n",
    "-5 & -6 & 11\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-70 & -79 & 148\\\\76 & 86 & -161\\\\-5 & -6 & 11\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-70  -79  148 ⎤\n",
       "⎢              ⎥\n",
       "⎢76   86   -161⎥\n",
       "⎢              ⎥\n",
       "⎣-5   -6    11 ⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Транспонированная матрица алгебраических дополнений\n",
    "# The transposed matrix of algebraic complements\n",
    "A.adjugate()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4) Вычислим обратную матрицу $\\large{A^{-1} = \\frac{A_S^T}{\\Delta_A}}$\n",
    "\n",
    "$A^{-1} = \n",
    "\\begin{pmatrix}\n",
    "70 & 79 & -148 \\\\\n",
    "-76 & -86 & 161 \\\\\n",
    "5 & 6 & -11\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}70 & 79 & -148\\\\-76 & -86 & 161\\\\5 & 6 & -11\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡70   79   -148⎤\n",
       "⎢              ⎥\n",
       "⎢-76  -86  161 ⎥\n",
       "⎢              ⎥\n",
       "⎣ 5    6   -11 ⎦"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.adjugate() / A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}70 & 79 & -148\\\\-76 & -86 & 161\\\\5 & 6 & -11\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡70   79   -148⎤\n",
       "⎢              ⎥\n",
       "⎢-76  -86  161 ⎥\n",
       "⎢              ⎥\n",
       "⎣ 5    6   -11 ⎦"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.inv()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4) Проверка $A * A^{-1} = E$:\n",
    "    \n",
    "$\\begin{pmatrix}\n",
    "20 & 19 & 9 \\\\\n",
    "31 & 30 & 22 \\\\\n",
    "26 & 25 & 16 \\\\\n",
    "\\end{pmatrix} *\n",
    "\\begin{pmatrix}\n",
    "70 & 79 & -148 \\\\\n",
    "-76 & -86 & 161 \\\\\n",
    "5 & 6 & -11\\\\\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "1& 0 & 0 \\\\\n",
    "0 & 1 & 0 \\\\\n",
    "0 & 0 & 1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "20 * 70 + 19 * (-76) + 9 * 5 = 1 &  20 * 79 + 19 * (-86) + 9 * 6 = 0 & 20 * (-148) + 19 * 161 + 9 * (-11) = 0 \\\\\n",
    "31 * 70 + 30 * (-76) + 22 * 5 = 0 &  31 * 79 + 30 * (-86) + 22 * 6 = 1 & 31 * (-148) + 30 * 161 + 22 * (-11) = 0 \\\\\n",
    "26 * 70 + 25 * (-76) + 16 * 5 = 0 &  26 * 79 + 25 * (-86) + 16 * 6 = 0 & 26 * (-148) + 25 * 161 + 16 * (-11) = 1 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 1, \\quad 0, \\quad 0\\right )$$"
      ],
      "text/plain": [
       "(1, 0, 0)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "20 * 70 + 19 * (-76) + 9 * 5, 20 * 79 + 19 * (-86) + 9 * 6, 20 * (-148) + 19 * 161 + 9 * (-11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 0, \\quad 1, \\quad 0\\right )$$"
      ],
      "text/plain": [
       "(0, 1, 0)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "31 * 70 + 30 * (-76) + 22 * 5,  31 * 79 + 30 * (-86) + 22 * 6, 31 * (-148) + 30 * 161 + 22 * (-11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 0, \\quad 0, \\quad 1\\right )$$"
      ],
      "text/plain": [
       "(0, 0, 1)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "26 * 70 + 25 * (-76) + 16 * 5,  26 * 79 + 25 * (-86) + 16 * 6,26 * (-148) + 25 * 161 + 16 * (-11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  1  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  0  1⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * A.inv()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет 3 / Standard Calculation 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "Решить систему линейных алгебраических уравнений двумя способами: по правилу Крамера и методом Гаусса\n",
    "        \n",
    "$\n",
    "\\begin{cases}\n",
    "8x_1 - 5x_2 - 2x_3 - 2x_4 = 35 \\\\\n",
    "-2x_1 + 4x_2 - 4x_3 + x_4 = -8 \\\\\n",
    "-2x_1 + x_2 + 4x_3 + 4x_4 = -11 \\\\\n",
    "4x_1 + x_2 + 3x_3 + 10x_4 = 10\\\\\n",
    "\\end{cases}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decisions:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Правило Крамера\n",
    "\n",
    "Матричный вид записи: Ax=b, где\n",
    "    \n",
    "$ A = \n",
    "\\begin{bmatrix}\n",
    "8 & -5 & -2 & -2  \\\\\n",
    "-2 & 4 & -4 & 1 \\\\\n",
    "-2 & 1 & 4 & 4 \\\\\n",
    "4 & 1 & 3 & 10\n",
    " \\end{bmatrix}, \\ \n",
    "b = \n",
    "\\begin{bmatrix}\n",
    "35 \\\\ -8 \\\\ -11 \\\\ 10\n",
    " \\end{bmatrix}\n",
    "$.\n",
    "\n",
    "Вычислим определитель матрицы системы:\n",
    "\n",
    "$ \\Delta = \n",
    "\\begin{vmatrix}\n",
    "8 & -5 & -2 & -2  \\\\\n",
    "-2 & 4 & -4 & 1 \\\\\n",
    "-2 & 1 & 4 & 4 \\\\\n",
    "4 & 1 & 3 & 10\n",
    " \\end{vmatrix}\n",
    "$.\n",
    "\n",
    "Вычтем из 2-ой строки 3-ю, из 1-ой - 4-ую, умноженную на 2:\n",
    "\n",
    "$ \\Delta = \n",
    "\\begin{vmatrix}\n",
    "0 & -7 & -8 & -22  \\\\\n",
    "0 & 3 & -8 & -3 \\\\\n",
    "-2 & 1 & 4 & 4 \\\\\n",
    "4 & 1 & 3 & 10\n",
    " \\end{vmatrix}\n",
    "$.\n",
    "\n",
    "Сложим 4-ую строку с 3-ей, умноженной на 2:\n",
    "\n",
    "$ \\Delta = \n",
    "\\begin{vmatrix}\n",
    "0 & -7 & -8 & -22  \\\\\n",
    "0 & 3 & -8 & -3 \\\\\n",
    "-2 & 1 & 4 & 4 \\\\\n",
    "0 & 3 & 11& 18\n",
    " \\end{vmatrix}\n",
    "$.\n",
    "\n",
    "Разложим по 1-му столбцу:\n",
    "\n",
    "$ \\Delta = -2 *\n",
    "\\begin{vmatrix}\n",
    " -7 & -8 & -22  \\\\\n",
    "3 & -8 & -3 \\\\\n",
    "3 & 11& 18\n",
    " \\end{vmatrix}\n",
    "$.\n",
    "\n",
    "Вычтем из 3-ей строки 2-ую:\n",
    "\n",
    "$ \\Delta = -2 *\n",
    "\\begin{vmatrix}\n",
    " -7 & -8 & -22  \\\\\n",
    "3 & -8 & -3 \\\\\n",
    "0 & 19 & 21\n",
    " \\end{vmatrix}\n",
    "$.\n",
    "\n",
    "К 1-ой строке прибавим 2-ую, умноженную на 2:\n",
    "\n",
    "$ \\Delta = -2 *\n",
    "\\begin{vmatrix}\n",
    " -1 & -24 & -28  \\\\\n",
    "3 & -8 & -3 \\\\\n",
    "0 & 19 & 21\n",
    " \\end{vmatrix}\n",
    "$.\n",
    "\n",
    "К 2-ой строке прибавим 1-ую, умноженную на 3:\n",
    "\n",
    "$ \\Delta = -2 *\n",
    "\\begin{vmatrix}\n",
    " -1 & -24 & -28  \\\\\n",
    "0 & -80 & -87 \\\\\n",
    "0 & 19 & 21\n",
    " \\end{vmatrix}\n",
    "$.\n",
    "\n",
    "Разложим по 1-му столбцу:\n",
    "    \n",
    "$ \\Delta = -2 * (-1) *\n",
    "\\begin{vmatrix}\n",
    "-80 & -87 \\\\\n",
    "19 & 21\n",
    " \\end{vmatrix} \n",
    " = 2 * (-80 * 21 - 19 * (-87)) = - 54$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Так как $\\Delta \\neq 0$ , то по теореме Крамера система совместна и имеет единственное решение. \n",
    "\n",
    "Вычислим вспомогательные определители. \n",
    "\n",
    "Определитель $\\Delta _i$  получим из определителя $\\Delta $ заменой его i-ого столбца столбцом свободных коэффициентов.\n",
    "\n",
    "Рассчитаем вспомогательные определители аналогично $\\Delta $ :\n",
    "\n",
    "$ \\Delta_1 = \n",
    "\\begin{vmatrix}\n",
    "35 & -5 & -2 & -2  \\\\\n",
    "-8 & 4 & -4 & 1 \\\\\n",
    "-11 & 1 & 4 & 4 \\\\\n",
    "10 & 1 & 3 & 10\n",
    " \\end{vmatrix} = -54\n",
    "\\large{\\implies x_1 = \\frac{\\Delta_1}{\\Delta} = -54 / (-54) = 1 }$.\n",
    "\n",
    "$ \\Delta_2 = \n",
    "\\begin{vmatrix}\n",
    "8 & 35 & -2 & -2  \\\\\n",
    "-2 & -8 & -4 & 1 \\\\\n",
    "-2 & -11 & 4 & 4 \\\\\n",
    "4 & 10 & 3 & 10\n",
    " \\end{vmatrix} = 270\n",
    "\\large{\\implies x_2 = \\frac{\\Delta_2}{\\Delta} = 270 / (-54) = -5 }$.\n",
    "\n",
    "$ \\Delta_3 = \n",
    "\\begin{vmatrix}\n",
    "8 & -5 & 35 & -2  \\\\\n",
    "-2 & 4 & -8 & 1 \\\\\n",
    "-2 & 1 & -11 & 4 \\\\\n",
    "4 & 1 & 10 & 10\n",
    " \\end{vmatrix} = 162\n",
    "\\large{\\implies x_3 = \\frac{\\Delta_3}{\\Delta} = 162 / (-54) = -3 }$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Подставим в последнее уравнение $x_1, x_2, x_3$:\n",
    "\n",
    "$4 * 1 + 1 * (-5) + 3 * (-3) + 10 * x_4 = 10 \\implies 10 * x_4 = 20 \\implies x_4 = 2 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}8 & -5 & -2 & -2\\\\-2 & 4 & -4 & 1\\\\-2 & 1 & 4 & 4\\\\4 & 1 & 3 & 10\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡8   -5  -2  -2⎤     ⎞\n",
       "⎜⎢              ⎥     ⎟\n",
       "⎜⎢-2  4   -4  1 ⎥     ⎟\n",
       "⎜⎢              ⎥, -54⎟\n",
       "⎜⎢-2  1   4   4 ⎥     ⎟\n",
       "⎜⎢              ⎥     ⎟\n",
       "⎝⎣4   1   3   10⎦     ⎠"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 1) \n",
    "A = sympy.Matrix([[8,-5,-2,-2],[-2,4,-4,1],[-2,1,4,4],[4,1,3,10]])\n",
    "A, A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & -7 & -8 & -22\\\\0 & 3 & -8 & -3\\\\-2 & 1 & 4 & 4\\\\4 & 1 & 3 & 10\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡0   -7  -8  -22⎤     ⎞\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎜⎢0   3   -8  -3 ⎥     ⎟\n",
       "⎜⎢               ⎥, -54⎟\n",
       "⎜⎢-2  1   4    4 ⎥     ⎟\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎝⎣4   1   3   10 ⎦     ⎠"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A= sympy.Matrix([[0,-7,-8,-22],[0,3,-8,-3],[-2,1,4,4],[4,1,3,10]])\n",
    "A, A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & -7 & -8 & -22\\\\0 & 3 & -8 & -3\\\\-2 & 1 & 4 & 4\\\\0 & 3 & 11 & 18\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡0   -7  -8  -22⎤     ⎞\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎜⎢0   3   -8  -3 ⎥     ⎟\n",
       "⎜⎢               ⎥, -54⎟\n",
       "⎜⎢-2  1   4    4 ⎥     ⎟\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎝⎣0   3   11  18 ⎦     ⎠"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A= sympy.Matrix([[0,-7,-8,-22],[0,3,-8,-3],[-2,1,4,4],[0,3,11,18]])\n",
    "A, A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAABLCAMAAABnT8qlAAAAPFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMAzUSZq1TvEGYy3Xa7iSJQQDB8rw0HegAAAAlwSFlzAAAOxAAADsQBlSsOGwAABxlJREFUeAHlnNea6ygMgLGNe9mzy/u/6yJhmpALSciZb+yLiUxk4AchipwRwl1VbcRR4SXdF39PmExVPlqBbeq5/BrLO6p10NfeGJzq19KgHoNUny1vtaBhtm1nG2VUW/iFl7fBqvi0L0jDh/FFNyS17n3aIX5rTFF1ydN8Qr/Oa3PfiLamqZYRsvIS3H0cv077V1ZQEl6H+GslG31NrdW8+Gzge3lXewObXJVuLi9hAR/HF3LBjP2fEYrdr0N8xNFdYxUvPge0sd6367m+xJHVadPyEj7xefxNkT6pgkoe4ptnOMfBgq3YxrfxOwX8Uhumlwrhi2WKatwGnS8O8fGRFUdn9PTBTa0WDdSkfobXl1glqUbhJdT8fO9rxDmsRBW2xin+RodNmA2RpVJzQ8yMqCS3k3PzTiqAL5bQfceNcYovvY9Iqp4krEpVB5NoomsSfE28VAK/DUe/xGFna3SGv7nOsdonn+26LSocVie6+1eT86teKoEvlHd2vYos2uH31eQvM+RX/1TCQtUHQGmVW06RB6g6fN04ei99ft6HksDH7NcQWoI4dX1T1FA2A/7TcCeTDK+MqdhgRCqw7IESarViQfrPEtn+GX6vbk97ot/Hibzr+kULNdrAWXhJ3xQxfuHWrr0K/f5p79fKWaeu18XVGa+33HV+I/bHrCdLL0ERZfAXa/2DtwMEcmM/xWuJaqoRpNQVrGOGaIYNvqbiNsGKGuZ8L6FOGfzZokjinE/wty5nGh9hi3B7oqzMhkr7Vi8VxB+t7+/IZHaCT3vsK/dlel8og02H/pnr+wouLaQQfmWMvo1n/VPXR2v2lftC+I1Z9zfWB1iWhxj/YCbxKl70PKb3RzPzd3YCfFjv98b3qXjN95je165fL8i2fQKwff8c/Alcf0uWvM/Br5Tejcx29eO6/yGeX+959SZjpdP+71r24K6+Z3ddDcx8MtnDpr2/rWuz3F68OzPKDHOY59Ky3lr2dKrShzaw94JrDDes2PFLsodN8Ht4iJyJYG4Xf7Cs22EOkxlT1lv406Sm1dKLKTyuWGHYV9f4MyyM+sRHXMCLzDCHyY4p6y38kFfMEf4MTr+ia9507NcdGH42fmaYw+AzZX0MfxyqsDUGwJ+u8bFiLUwTWVdmmCPIOy7rY/irIPj6sH+KAx66EsnYh4rVUy49eNX8MAdT1nv4wzw35lB33mJ8HbAWOqBGuRj8cZbSHQxDDe9d+WEOnW9S1lv4EGfbcGe/ac6o9xFf3cHXtYriIvfoXwhzmIzjst7CxwwxogkHqRF+Db2v6H6XN34wEzd/8PQ0bnER5jCZ0IdMalRWDj6fXaM3NwMsfSJ8/Q5Tr/H/ITR/1L9xyoh2vyTtFGsld/lhDp0FU9Z/5CwyKegkYcGoVKPGDQ+cI/w/Bp8e3tKxrzfG0O+5+PlhDl0IV1ZO79OW6DCEoV+OqiVcqpMYTEA13ftg01f4osPZMjJIWgx3nxvmwDyYsgj+xRCMa2JgOxuaM7nvKjj2p2vXh8u3eC6OC+HvMsMcJhOmrBg/b/WJb+LMLsoabe/u4otahyte2PLkhTlsnyRlxfjCb2D4Vo9T9UtistkNptHxk8UHnRAfN/3RI3TsR1/+hRuCL9os8z+usI69g0fz7WFUfzq+d17HaHe+GQw+ze6H4/e0u+6gcjq445PXG17u2S+mEeOna/SXa4L7fTzyibL44b0f1fWdmwbmgTmaDCC7p+Cj3bfJMcZT8NHp76GuwIqego/BzT3U9UB8E9zEUFdA/5Sxv++uMNT1QPx91Eu650nHfhp6CJsrlkcfU3BSrJHe2Wf4uAiZ99PHX0sZjM+f6bonwWdCDwcl9lJvK3BJ7qUDVZccaPJxkUL4qznidm942Qol+Ezoweqmn7M7E/NSqhWn7JoHcZFC+Pb4hgb4E3wm9BBXP7zz0F4Kv+fkXfMgLlIIf9r7aSI/bErwscZ3jzs8tJc45DBt1zyIi5TB7+1LvSvxfSz+7TCHh/ZSiMrJVpOPi5TBb21ku7bCXjMGPwk9cBAmzaLAbuLuuYTTZOMiZfClO+OEH48FF4Ovv41DD4E6ER3KC/h8XKQMvn+VVx+Dhww8/nWYw+TxBv5BXKQI/uhNPvhlAyAk+EzoIWyuWH4D/yAuUgQfJ/O95rH1U3wu9BAjh3ev4x/FRYrgV8GAn6M3+il+XpjjdXxxEBcpga9/pB/0mR8IOjHBZ0IPwbNEfAP/IC5SAj/+6d4azlEJfkaYQy6dmiScxXqJtE9y6zX5uEgBfPKTrD60hRQ/qfFXEwrgz2Sh602WMf6vwqaFFcDvwvf7oMTOu4Lf3/thZ5vmbr05/Hr8nmxyoAX8P3T49fge1XQ9/nVvrv12/NqF+wN6MdrUX/5vi8TEv5xd76999/jPgn7Ovy0yZ2BhT70l9yH9/x9HSs6kj/kDAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-7 & -8 & -22\\\\3 & -8 & -3\\\\3 & 11 & 18\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡-7  -8  -22⎤     ⎞\n",
       "⎜⎢           ⎥     ⎟\n",
       "⎜⎢3   -8  -3 ⎥, -54⎟\n",
       "⎜⎢           ⎥     ⎟\n",
       "⎝⎣3   11  18 ⎦     ⎠"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A11= sympy.Matrix([[-7,-8,-22],[3,-8,-3],[3,11,18]])\n",
    "A11, -2 * A11.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-7 & -8 & -22\\\\3 & -8 & -3\\\\0 & 19 & 21\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡-7  -8  -22⎤     ⎞\n",
       "⎜⎢           ⎥     ⎟\n",
       "⎜⎢3   -8  -3 ⎥, -54⎟\n",
       "⎜⎢           ⎥     ⎟\n",
       "⎝⎣0   19  21 ⎦     ⎠"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A11= sympy.Matrix([[-7,-8,-22],[3,-8,-3],[0,19,21]])\n",
    "A11, -2 * A11.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-1 & -24 & -28\\\\0 & -80 & -87\\\\0 & 19 & 21\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡-1  -24  -28⎤     ⎞\n",
       "⎜⎢            ⎥     ⎟\n",
       "⎜⎢0   -80  -87⎥, -54⎟\n",
       "⎜⎢            ⎥     ⎟\n",
       "⎝⎣0   19   21 ⎦     ⎠"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A11= sympy.Matrix([[-1,-24,-28],[0,-80,-87],[0,19,21]])\n",
    "A11, -2 * A11.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-80 & -87\\\\19 & 21\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡-80  -87⎤     ⎞\n",
       "⎜⎢        ⎥, -54⎟\n",
       "⎝⎣19   21 ⎦     ⎠"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A111= sympy.Matrix([[-80,-87],[19,21]])\n",
    "A111, -2 * (-1) *  A111.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAS0AAABkCAMAAADkHZbMAAAAQlBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACGKEuzAAAAFXRSTlMAzUSZq1TvEGYy3Xa7iSJsYK9AMHz2s7cdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJSklEQVR4Ae2d2bqkKAyAcd+d6Z7h/V91WASSgAjVUm1/Y12cQkgw/AVIADmM2U9V6+DE1ae1CX9OYNam32nwPg+h7BqDZ+LbKD4Hu5DoY+Ok3WPLb7VvM1xgrl1vGE58hwkoPMmr4TwdCYOLT/VAFsnB8V5arB+9Ww8uzqe1b1uzqrrW86qaw3XTyxJGZOrtTVOtijDM5DSMxe+mVfu1p62sLR6toRFpI+/E33nm82YqodW4DuTp7bLybzy1LyDid9Ni7UoKOAHTPFqLBDVwyTPUhklWwcs8vVb9Hn0fzMqPJOK309pVRQH3rVzVYh6tupe/8hdp9Vziav0WACwGQSJ+Oy22zuBujHWgavm0lGjHZWeXV0fcPfL0WmVdyxN7LiJ+P62JL64ojFUQnle3pGQ9qydDOy5Lk1gIeINP9Oa8Z5sVv58WW2GngNkFaE1L2ypGq2giO6yIEEkk/IEeNiqSt05y4gVodbDnalU/YQwK0BJJQIE0Y6N3+Z2nN8sncfrHiRegpTttbczA0SMyTIvpvlRqNJfd7yAGZfZjG+6pXki8OYd1IV6CFuhD9VDK/owerUk3QlEbV/XobFK7X5tlvt54DsvmCgJQvAStmm/mbitqiN4zceAqfRW0+uNhZTRTv7P1OmncnuxhIfEStBg3/fzA4RORebRYr57/siVqwD1quCnEcvUmpbCkOg1YvAit1bSn0dUyVXKvJY5y8KDGW7tsH4v1v1NAKZlMvX1uxEcPoxLuQcSL0FoMpJaMCDxarBa2a69a+K9tk/qTg4Lm6VV6mgr4FyArP0jEi9CalCsj7t2TUaBPyzfwyTFFaDGuKdFuy++3nowmYFsZWpVugR0ebQV6+YBFT44qQ6vRvmJj+i9D4G2JhgT8HrkaGVTQB5LpLy1IyYQnPeLqzUjCxSePCo3Ko77LtEQxRpel1CN1UN63bgEYLsild7wfj0YX/dJyLEBolg/Fjvg9b78FCMFgJSePFzNItSlv3bIoYKCVY4eNDrfeZyJk5MKNHEK0ehzhYq9oDduyNalLfTLbKW+qyliSpDYFVoLLPBN1tVo5LcxFS1TirVyKTfzMeQs+JtdrtaFtKjw1p3SL0RJefpVJS03fsCF1fkB0jNfFNoDgd5ra8j1ai3wcVtTxuWiJm5oKTKc1jdUndStR7Yu0RklrzqRVc7na1ajVRVgXzsIb+4hWotpXaYnJ5hmvw4pCX/RbLedLk9xtLftHtFLVvkirk8P4Xq3Yw5pxQUs8HHiV6knuog5+ULeS1b5Ni+fS6rZ95WRyGsJGYbka8QGtZLUv0qpl3eL8L1Q+xn7yf0AMXepUK3cdT1vJGGUdvKbl3SNNTeQdovWvnhMGZbgl+FNOP3D+N8kM0yKJTGPytjRRMXW9q60p17SIcoba92nRHjvabw3HD9emPBTrVn5436rlQcLk/DJDLUSr0OhUbPWWdSuLFut1B7+m9vPiOfLJeCtR7em06kouJ45o+9d5NZEpntseFzepSWp301I7PsJ7tj+qW2xqxVpsulctXDm+ZrA9YCWptWvP55Zm/istEe+9Ro79Z7TMb//Y71+hhfdeIw/3peX94qiLxY79SytKizj2L60oLeLYv7R8Wm7PNnXsX1oeLbf32nPsX1oeLRWh9l57jr2i1fygo/Jh/GA7W/i+vyVW1orED3XntZrce+3PBww/fsg5iDzPJ9GO3yj2C+Mtu/c64Ni/LZH+pnbvdcCx/1/QyupV9FyJ3bON5gM+oOVWO12I/kD4OlXOaiHnzMaeBnxx1BL1a5Wn2iSB7L1Gjn0uLbfa6ULkduQyVQ6qIecMJoTDvjiiJd7WzaldcO81cexzaQlz3QyJC4VLYWJT5ay8X3yTFPrGvpySwLRYl0MrdIsj7om0iHMWsV4lhcQJrbx52/MbPpEWcc7OjdcpIXFMa6DTX1dZnqU/kBZ1zs5MP+KD4phW+lD14l7Po+U5Z/EShMUxrXgOGanPo+U5Z/HShMV/E62AF+WecC4UL1GqnMzFd86ieZ+Il6VFVyRi64mu7C4ULREYc8TlRGrAOYvpnImXpZXjVTtGLhQrERyhxeVEasA5i+mciZejJffYPIaWRoOcsxitM/FCtNSukTlrj42rUS4UL1GqnM0FOWc29jTgiz+FllvtdKHTYqiEVDmbC3HObPxJICRekpZ6xQDZEuvlkeBDLwrREruwGFtzd1I+lJEzqxCtUdOibudbtxx5EFI7wNvM/fJA/6HBQnVrky9EqddXULnfuoVwmItGPn0X7xH80jKA0LdqhN37xh2CcnqhHofH29VA6K1bAIYLqrf1j7erXezVuxhA8pnBQr28fltfvV0Ny/3WLUjDhI9ztdTb1SZOfr+0IA0TPnqslrrVIVp2DXXbloZuMzEZBr+tZjCVRLojjklC4NJmTE0q0xJH/TRc6PDUowXWUCsxvTPMybiAZqC8XhQ44thLwxEgY8+kMrQ2/WaiPVnK2OPREgnHvMuoToHTL3Qa8YvvnBkbcMTxRa4xk8rQkgcByg89PiNCa1V7frvA67kqp9CfHFrgiONQVjjuyNg3qQyt+Sj0TA7kjNDSjbb2pltxOdBVDi2lqI84RnkEL46MfZOK0BrMMYsb6ebPaQ3GtIwlzVxaxxHHQUAoUmccMKkIrc4cBFGbwGHMOa3d9HQZq+V5tOwRxwhM8EJnHDCpCK3WtidykHsCLTolFiyOjsyjJXTAEceRbM2Dx9JyJhWh5Q5XJGfjntMadN2qvenWSLGyaYEjjiPZGloBk0rQmlz7A8fqSvvOaTHdb6X++qqwObTcEcdRTjrxyNg3qQQtNbY5jMJNMUJrVUdnjIVGEOCI43RavkklaFXm4GFh2IJewY/QGpVSk34si2kwCaUXIu6I4wT5o275JhWgJf7FEbDItUoRGaHFZun59PYMdJDFWTCnJbojjs9yA/EmY8+kArTwAfwbbFs+LbeGKo6TWvR/MQB2R4JOMyLkktwRxy4uHHIZeybdT2tAtUn8+xlQ03xaYYOfGns/rYV4O6ZaSwIvLVoPenpEWe8q10uL0IJVSSd1rrK9tDCtgfjRMtX9s62XFqblyIB4/e+NRMRLC1AROxaDBxxNJlZsdZYf9LI/0n/uRYH/zclmui9XF78+ptsH9R8u/9z/zanPgbzrJx0grP8AkLRwMQw3hsEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}35 & -5 & -2 & -2\\\\-8 & 4 & -4 & 1\\\\-11 & 1 & 4 & 4\\\\10 & 1 & 3 & 10\\end{matrix}\\right], \\quad -54\\right )$$"
      ],
      "text/plain": [
       "⎛⎡35   -5  -2  -2⎤     ⎞\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎜⎢-8   4   -4  1 ⎥     ⎟\n",
       "⎜⎢               ⎥, -54⎟\n",
       "⎜⎢-11  1   4   4 ⎥     ⎟\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎝⎣10   1   3   10⎦     ⎠"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 2)\n",
    "A_1 = sympy.Matrix([[35,-5,-2,-2],[-8,4,-4,1],[-11,1,4,4],[10,1,3,10]])\n",
    "A_1, A_1.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOBAMAAADgeEClAAAAIVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADdcGRXAAAACnRSTlMAMt0Qq5nNdrvvxbMB0AAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABtJREFUCB1jYGBUZmAwCVvMwMDAThbB0rnIHQBAqQoWQ8kakgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x1 = A_1.det() / A.det()\n",
    "x1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}8 & 35 & -2 & -2\\\\-2 & -8 & -4 & 1\\\\-2 & -11 & 4 & 4\\\\4 & 10 & 3 & 10\\end{matrix}\\right], \\quad 270\\right )$$"
      ],
      "text/plain": [
       "⎛⎡8   35   -2  -2⎤     ⎞\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎜⎢-2  -8   -4  1 ⎥     ⎟\n",
       "⎜⎢               ⎥, 270⎟\n",
       "⎜⎢-2  -11  4   4 ⎥     ⎟\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎝⎣4   10   3   10⎦     ⎠"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_2 = sympy.Matrix([[8,35,-2,-2],[-2,-8,-4,1],[-2,-11,4,4],[4,10,3,10]])\n",
    "A_2, A_2.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAPBAMAAADjSHnWAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEM3dMiLviatmdpm7RDzvOQYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABcSURBVAgdY2CAAhYGBlEYm4GB7ZGyIYLHbGcugOBxIJhAFl7erNUOCMU8AqxvETwg6zADg5AxCJiAhPcFwCUPAXkOcF4yA0MfnMNQwMCQg+BFM3AlInisu7sFGAAwbw1VKWmJxQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$-5$$"
      ],
      "text/plain": [
       "-5"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x2 = A_2.det() / A.det()\n",
    "x2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}8 & -5 & 35 & -2\\\\-2 & 4 & -8 & 1\\\\-2 & 1 & -11 & 4\\\\4 & 1 & 10 & 10\\end{matrix}\\right], \\quad 162\\right )$$"
      ],
      "text/plain": [
       "⎛⎡8   -5  35   -2⎤     ⎞\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎜⎢-2  4   -8   1 ⎥     ⎟\n",
       "⎜⎢               ⎥, 162⎟\n",
       "⎜⎢-2  1   -11  4 ⎥     ⎟\n",
       "⎜⎢               ⎥     ⎟\n",
       "⎝⎣4   1   10   10⎦     ⎠"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_3 = sympy.Matrix([[8,-5,35,-2],[-2,4,-8,1],[-2,1,-11,4],[4,1,10,10]])\n",
    "A_3, A_3.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKJmbtEZlR276s0Yu9/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXElEQVQIHWNggALWtIoJMDYDgzgD128E7/wGxk8I3twJDEg8BgbuXwg5BobJC5B4Pu8ckHgM3EAzhYxBwAQkfF8ALunCwNC/AcZj/C+AxGO4h6JyFYp9nOVAVwMANtkW44cnmdIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-3$$"
      ],
      "text/plain": [
       "-3"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x3 = A_3.det() / A.det()\n",
    "x3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}8 & -5 & -2 & 35\\\\-2 & 4 & -4 & -8\\\\-2 & 1 & 4 & -11\\\\4 & 1 & 3 & 10\\end{matrix}\\right], \\quad -108\\right )$$"
      ],
      "text/plain": [
       "⎛⎡8   -5  -2  35 ⎤      ⎞\n",
       "⎜⎢               ⎥      ⎟\n",
       "⎜⎢-2  4   -4  -8 ⎥      ⎟\n",
       "⎜⎢               ⎥, -108⎟\n",
       "⎜⎢-2  1   4   -11⎥      ⎟\n",
       "⎜⎢               ⎥      ⎟\n",
       "⎝⎣4   1   3   10 ⎦      ⎠"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_4 = sympy.Matrix([[8,-5,-2,35],[-2,4,-4,-8],[-2,1,4,-11],[4,1,3,10]])\n",
    "A_4, A_4.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNuzKriUQQ7912ZlSugimNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAT0lEQVQIHWNgVDIJYGAQY2D/wcCQycCwioFh/QSGfAGGbAaGfKAEA8N+EMH6B0RyK4BIJRDBqwAinzMwCjCwFjDwTGCQVlICarH///8zAwAXpw31OpgbMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x4 = A_4.det() / A.det()\n",
    "x4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Метод Гаусса\n",
    "\n",
    "Запишем расширенную матрицу:\n",
    "\n",
    "$\n",
    " \\left(\\begin{array}{cccc|c}\n",
    "8 & -5 & -2 & - 2 &  35 \\\\\n",
    "-2 & 4 & -4 & 1 &  -8 \\\\\n",
    "-2 & 1 & 4 & 4 & -11 \\\\\n",
    "4 & 1 & 3 & 10 & 10 \n",
    " \\end{array}\\right)\n",
    " $\n",
    " \n",
    "Умножим 2-ую и 3-ю строки на 4, 4-ую - на -2 и сложим с 1-ой:\n",
    " \n",
    " $\n",
    " \\left(\\begin{array}{cccc|c}\n",
    "8 & -5 & -2 & - 2 &  35 \\\\\n",
    "0 & 11 & -18 & 2 &  3 \\\\\n",
    "0 & -1 & 14 & 14 & -9 \\\\\n",
    "0 & -7 & -8 & -22 & 15\n",
    " \\end{array}\\right)\n",
    " $\n",
    " \n",
    " Умножим 3-ю строчку на 11 и сложим со 2-й; умножим 2-ую на 7, 4-ую - на 11 и сложим:\n",
    " \n",
    " $\n",
    " \\left(\\begin{array}{cccc|c}\n",
    "8 & -5 & -2 & - 2 &  35 \\\\\n",
    "0 & 11 & -18 & 2 &  3 \\\\\n",
    "0 & 0 & 136 & 156 & -96 \\\\\n",
    "0 & 0 & -214 & -228 & 186\n",
    " \\end{array}\\right) \n",
    " $\n",
    " \n",
    " Поделим 3-ю строчку на 4, 4-ую - на 2:\n",
    " \n",
    " $\n",
    " \\left(\\begin{array}{cccc|c}\n",
    "8 & -5 & -2 & - 2 &  35 \\\\\n",
    "0 & 11 & -18 & 2 &  3 \\\\\n",
    "0 & 0 & 34 & 39 & -24 \\\\\n",
    "0 & 0 & -107 & -114 & 93\n",
    " \\end{array}\\right) \n",
    " $\n",
    "\n",
    "Умножим 3-ю строчку на 107, 4-ую - на 34 и сложим:\n",
    "\n",
    " $\n",
    " \\left(\\begin{array}{cccc|c}\n",
    "8 & -5 & -2 & - 2 &  35 \\\\\n",
    "0 & 11 & -18 & 2 &  3 \\\\\n",
    "0 & 0 & 34 & 39 & -24 \\\\\n",
    "0 & 0 & 0 & 297 & 594\n",
    " \\end{array}\\right) \n",
    " $\n",
    " \n",
    " Поделим 4-ю на 297:\n",
    " \n",
    "  $\n",
    " \\left(\\begin{array}{cccc|c}\n",
    "8 & -5 & -2 & - 2 &  35 \\\\\n",
    "0 & 11 & -18 & 2 &  3 \\\\\n",
    "0 & 0 & 34 & 39 & -24 \\\\\n",
    "0 & 0 & 0 & 1 & 2\n",
    " \\end{array}\\right) \n",
    " $\n",
    " \n",
    "4-ая строка означает: $x_4 = 2 $.\n",
    " \n",
    "Подставим в 3-е уравнение:\n",
    " \n",
    " $x_4 = 2 \\implies 34x_3 + 39*2 = -24 \\implies  34x_3 = -102 \\implies x_3 = -3$\n",
    " \n",
    " Подставим в 2-е уравнение:\n",
    " \n",
    " $x_4 = 2 , x_3 = -3 \\implies 11x_2 - 18 * (-3) + 2 * 2 = 3 \\implies 11x_2 = -55 \\implies x_2 = -5$\n",
    " \n",
    " Подставим в 1-ое уравнение:\n",
    " \n",
    " $x_4 = 2 , x_3 = -3, x_2 = -5 \\implies 8x_1 - 5 * (-5) - 2 * (-3) - 2 * 2 = 35 \\implies 8x_1 = 8 \\implies x_1 = 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ответ / Answer: (1, -5, -3, 2)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ a : 1, \\quad b : -5, \\quad c : -3, \\quad d : 2\\right \\}$$"
      ],
      "text/plain": [
       "{a: 1, b: -5, c: -3, d: 2}"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import a,b,c,d\n",
    "system = sympy.Matrix([[8,-5,-2, -2,35],[-2,4, -4, 1,-8] ,[-2,1,4,4,-11],[4,1,3,10,10]])\n",
    "sympy.solve_linear_system(system, a,b,c,d)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}