lesson4.ipynb

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   "source": [
    "import numpy as np"
   ],
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   "source": [
    "__1.__ Вычислить определитель:\n",
    "\n",
    "   a)\n",
    "\n",
    "$$\\begin{vmatrix}\n",
    "sinx & -cosx\\\\\n",
    "cosx & sinx\n",
    "\\end{vmatrix};$$\n",
    "\n",
    "   б)\n",
    "\n",
    "$$\\begin{vmatrix}\n",
    "4 & 2 & 3\\\\\n",
    "0 & 5 & 1\\\\\n",
    "0 & 0 & 9\n",
    "\\end{vmatrix};$$\n",
    "\n",
    "   в)\n",
    "\n",
    "$$\\begin{vmatrix}\n",
    "1 & 2 & 3\\\\\n",
    "4 & 5 & 6\\\\\n",
    "7 & 8 & 9\n",
    "\\end{vmatrix}.$$"
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   "source": [
    "Вариант а:\n",
    "\n",
    "$det = sinx * sinx - (cosx * -cosx) = sin^2x + cos^2x = 1$"
   ],
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   "cell_type": "markdown",
   "source": [
    "Вариант б:"
   ],
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   "source": [
    "a1_2 = np.array([[4,2,3], [0,5,1], [0,0,9]])\n",
    "det1_2 = np.linalg.det(a1_2)\n",
    "det1_2"
   ],
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    "Вариант в:"
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     "execution_count": 28,
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   "source": [
    "a1_3 = np.array([[1,2,3], [4,5,6], [7,8,9]])\n",
    "det1_3 = np.linalg.det(a1_3)\n",
    "det1_3"
   ],
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   "source": [
    "__2.__ Определитель матрицы $A$ равен $4$. Найти:\n",
    "\n",
    "   а) $det(A^{2})$;\n",
    "\n",
    "   б) $det(A^{T})$;\n",
    "\n",
    "   в) $det(2A)$."
   ],
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   "source": [
    "Вариант а:\n",
    "\n",
    "$det(A^{2}) = det(AA) = det(A)*det(a) = 4 * 4 = 16$"
   ],
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   "source": [
    "Вариант б:\n",
    "\n",
    "$det(A^{T}) = det(A) = 4$"
   ],
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   "source": [
    "Вариант в:\n",
    "\n",
    "$det(2A) = det(A) * 2^n = 4 * 2^n$, n - кол-во строк в матрице"
   ],
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   "source": [
    "__3.__  Доказать, что матрица\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "-2 & 7 & -3\\\\\n",
    "4 & -14 & 6\\\\\n",
    "-3 & 7 & 13\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "вырожденная."
   ],
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   "source": [
    "a3 = np.array([[-2,7,-3], [4,-14,6], [-3,7,13]])\n",
    "det3 = np.linalg.det(a3)\n",
    "det3"
   ],
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   "cell_type": "markdown",
   "source": [
    "определитель матрицы равен нулю, следовательно, матрица вырожденная"
   ],
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   "source": [
    "__4.__ Найти ранг матрицы:\n",
    "\n",
    "   а) $\\begin{pmatrix}\n",
    "1 & 2 & 3\\\\\n",
    "1 & 1 & 1\\\\\n",
    "2 & 3 & 4\n",
    "\\end{pmatrix};$\n",
    "\n",
    "   б) $\\begin{pmatrix}\n",
    "0 & 0 & 2 & 1\\\\\n",
    "0 & 0 & 2 & 2\\\\\n",
    "0 & 0 & 4 & 3\\\\\n",
    "2 & 3 & 5 & 6\n",
    "\\end{pmatrix}.$"
   ],
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   "cell_type": "markdown",
   "source": [
    "Вариант а:"
   ],
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     "name": "#%% md\n"
    }
   }
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  {
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     "execution_count": 30,
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   "source": [
    "a4_1 = np.array([[1,2,3], [1,1,1], [2,3,4]])\n",
    "result4_1 = np.linalg.matrix_rank(a4_1)\n",
    "result4_1"
   ],
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   "execution_count": 31,
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     "execution_count": 31,
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   "source": [
    "a4_2 = np.array([[0,0,2,1], [0,0,2,2], [0,0,4,3], [2,3,5,6]])\n",
    "result4_2 = np.linalg.matrix_rank(a4_2)\n",
    "result4_2"
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