lesson6.ipynb

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    "__1.__ Решить систему уравнений методом Гаусса:\n",
    "\n",
    "$$\\begin{cases}\n",
    "x_{1}+x_{2}-x_{3}-2x_{4}=0, \\\\\n",
    "2x_{1}+x_{2}-x_{3}+x_{4}=-2, \\\\\n",
    "x_{1}+x_{2}-3x_{3}+x_{4}=4.\n",
    "\\end{cases}$$"
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    "Преобразуем расширенную матрицу системы путем элементарных преобразований:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 1 & -1 & -2 \\\\\n",
    "2 & 1 & -1 & 1 \\\\\n",
    "1 & 1 & -3 & 1\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0\\\\\n",
    "-2\\\\\n",
    "4\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из второй строки первую, умноженную на 2:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 1 & -1 & -2\\\\\n",
    "0 & -1 & 1 &5\\\\\n",
    "1 & 1 & -3 & 1\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0\\\\\n",
    "-2\\\\\n",
    "4\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из третьей строки первую:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 1 & -1 & -2\\\\\n",
    "0 & -1 & 1 & 5\\\\\n",
    "0 & 0 & -2 & 3\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0\\\\\n",
    "-2\\\\\n",
    "4\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Система имеет бесконечное кол-во решений.\n",
    "\n",
    "Поделим третью строку на 3:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 1 & -1 & -2\\\\\n",
    "0 & -1 & 1 & 5\\\\\n",
    "0 & 0 & -\\frac{2}{3} & 1\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0\\\\\n",
    "-2\\\\\n",
    "\\frac{4}{3}\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Оставшаяся матрица соответствует системе:\n",
    "\n",
    "$$\\begin{cases}\n",
    "x_{1} + x_{2} - x_{3} -2 \\cdot x_{4} = 0, \\\\\n",
    "-x_{2} + x_{3} + 5 \\cdot x_{4} = -2, \\\\\n",
    "-\\frac{2}{3} \\cdot x_{3} + x_{4} = \\frac{4}{3}\n",
    "\\end{cases}$$\n",
    "\n",
    "Бесконечное множество решений принято записывать в виде _общего решения_.\n",
    "\n",
    "Последнее уравнение содержит две неизвестных. В данном случае можно рассматривать в качестве _свободного_ параметра $x_{4}=c$. Тогда, выражая остальные переменные через $c$, получим:\n",
    "\n",
    "$x_{4}=c,$\n",
    "\n",
    "$x_{3}=\\frac{4-3c}{-2} \\;$\n",
    "\n",
    "$x_{2}=2 + 5 \\cdot c + \\frac{4-3c}{-2},$\n",
    "\n",
    "$x_{1}=-2 -3 \\cdot c.$\n",
    "\n",
    "Таковым будет _общее решение системы_. Подставляя произвольные числа вместо $c$, мы получим _частное решение_. Например, при $c=0$:\n",
    "\n",
    "$x_{1}=-2,$\n",
    "$x_{2}=0,$\n",
    "$x_{3}=-2,$\n",
    "$x_{4}=0.$\n",
    "\n",
    "Параметр $c$ может принимать бесконечное количество значений, при которых уравнения в системе будут обращаться в тождества."
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    "__2.__ Проверить на совместность и выяснить, сколько решений будет иметь система линейных уравнений:\n",
    "\n",
    "   а) $\\begin{cases}\n",
    "3x_{1}-x_{2}+x_{3}=4, \\\\\n",
    "2x_{1}-5x_{2}-3x_{3}=-17, \\\\\n",
    "x_{1}+x_{2}-x_{3}=0;\n",
    "\\end{cases}$"
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    "**Решение 2а):**\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "3 & -1 & 1 \\\\\n",
    "2 & -5 & -3 \\\\\n",
    "1 & 1 & -1\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "4\\\\\n",
    "-17\\\\\n",
    "0\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Поделим первую строку на 3:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
    "2 & -5 & -3 \\\\\n",
    "1 & 1 & -1\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "\\frac{4}{3}\\\\\n",
    "-17\\\\\n",
    "0\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из второй строки первую, умноженную на 2:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
    "0 & -\\frac{13}{3} & -\\frac{11}{3} \\\\\n",
    "1 & 1 & -1\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "\\frac{4}{3}\\\\\n",
    "-\\frac{59}{3}\\\\\n",
    "0\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из третьей строки первую:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
    "0 & -\\frac{13}{3} & -\\frac{11}{3} \\\\\n",
    "0 & \\frac{4}{3} & -\\frac{4}{3}\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "\\frac{4}{3}\\\\\n",
    "-\\frac{59}{3}\\\\\n",
    "-\\frac{4}{3}\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Умножим вторую строку на $-\\frac{3}{13}$, и отнимем вторую строку из третьей:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
    "0 & 1 & \\frac{11}{13} \\\\\n",
    "0 & 0 & \\frac{24}{13}\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "\\frac{4}{3}\\\\\n",
    "\\frac{59}{13}\\\\\n",
    "\\frac{72}{13}\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Упростим третью строку, путем умножения на 13 и деления полученных значений на 24:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
    "0 & 1 & \\frac{11}{13} \\\\\n",
    "0 & 0 & 1\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "\\frac{4}{3}\\\\\n",
    "\\frac{59}{13}\\\\\n",
    "3\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "**$rangA=rang(A|B)=n$, следовательно, система совместная и определенная**"
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    "б) $\\begin{cases}\n",
    "2x_{1}-4x_{2}+6x_{3}=1, \\\\\n",
    "x_{1}-2x_{2}+3x_{3}=-2, \\\\\n",
    "3x_{1}-6x_{2}+9x_{3}=5;\n",
    "\\end{cases}$\n",
    "\n",
    "**Решение 2б):**\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "2 & -4 & 6 \\\\\n",
    "1 & -2 & 3 \\\\\n",
    "3 & 6 & -9\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "1\\\\\n",
    "-2\\\\\n",
    "5\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Поделим первую строку на 2:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -2 & 3 \\\\\n",
    "1 & -2 & 3 \\\\\n",
    "3 & 6 & -9\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0.5\\\\\n",
    "-2\\\\\n",
    "5\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из второй строки первую:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -2 & 3 \\\\\n",
    "0 & 0 & 0 \\\\\n",
    "3 & 6 & -9\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0.5\\\\\n",
    "-2.5\\\\\n",
    "5\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из третьей строки первую, умноженную на 3:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -2 & 3 \\\\\n",
    "0 & 0 & 0 \\\\\n",
    "0 & 0 & 0\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0.5\\\\\n",
    "-2.5\\\\\n",
    "3.5\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Поделим вторую строку на -2.5:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -2 & 3 \\\\\n",
    "0 & 0 & 0 \\\\\n",
    "0 & 0 & 0\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0.5\\\\\n",
    "1\\\\\n",
    "3.5\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из третьей строки вторую, умноженную на 3.5:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & -2 & 3 \\\\\n",
    "0 & 0 & 0 \\\\\n",
    "0 & 0 & 0\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "0.5\\\\\n",
    "1\\\\\n",
    "0\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "**$rangA$ не равен $rang(A|B)$, следовательно, система несовместная**"
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    "в) $\\begin{cases}\n",
    "x_{1}+2x_{2}+5x_{3}=4, \\\\\n",
    "3x_{1}+x_{2}-8x_{3}=-2.\n",
    "\\end{cases}$\n",
    "\n",
    "**Решение 2в):**\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 2 & 5 \\\\\n",
    "3 & 1 & -8\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "4\\\\\n",
    "-2\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из второй строки первую, умноженную на 3:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 2 & 5 \\\\\n",
    "0 & -5 & -23\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "4\\\\\n",
    "-14\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Разделим вторую строку на -5:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 2 & 5 \\\\\n",
    "0 & 1 & \\frac{23}{5}\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "4\\\\\n",
    "\\frac{14}{5}\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "**$rangA=rang(A|B)<n$, следовательно, система совместная и неопределенная**"
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   "source": [
    "__3.__ Проверить на совместность и выяснить, сколько решений будет иметь система линейных уравнений, заданная расширенной матрицей\n",
    "\n",
    "$$\\tilde{A}=\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 3 & -2 & 4\\\\\n",
    "0 & 5 & 0 & 1\\\\\n",
    "0 & 0 & 3 & 0\\\\\n",
    "0 & 0 & 0 & 2\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "3\\\\\n",
    "2\\\\\n",
    "4\\\\\n",
    "1\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}.$$\n",
    "\n",
    "**$rang(A)=rang(A|B)=n$, следовательно, система совместная и определенная**"
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    "__4.__ Дана система линейных уравнений, заданная расширенной матрицей\n",
    "\n",
    "$$\\tilde{A}=\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 2 & 3\\\\\n",
    "4 & 5 & 6\\\\\n",
    "7 & 8 & 9\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "a\\\\\n",
    "b\\\\\n",
    "c\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}.$$\n",
    "\n",
    "Найти соотношение между параметрами $a$, $b$ и $c$, при которых система является несовместной."
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    "**Решение:**\n",
    "\n",
    "Отнимем из второй строки первую, умноженную на 4:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 2 & 3\\\\\n",
    "0 & -3 & -6\\\\\n",
    "7 & 8 & 9\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "a\\\\\n",
    "b-4 \\cdot a\\\\\n",
    "c\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из третьей строки первую, умноженную на 7:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 2 & 3\\\\\n",
    "0 & -3 & -6\\\\\n",
    "0 & -6 & -12\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "a\\\\\n",
    "b-4 \\cdot a\\\\\n",
    "c-7 \\cdot a\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "Отнимем из третьей строки вторую, умноженную на -2:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\left.\\begin{matrix}\n",
    "1 & 2 & 3\\\\\n",
    "0 & -3 & -6\\\\\n",
    "0 & 0 & 0\n",
    "\\end{matrix}\\right|\n",
    "\\begin{matrix}\n",
    "a\\\\\n",
    "b-4 \\cdot a\\\\\n",
    "c-7 \\cdot a + 2\\cdot b -8\\cdot a\n",
    "\\end{matrix}\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "**При $(-15 \\cdot a + 2\\cdot b + c)\\neq0$ система является несовместной**"
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    "__5.__ Решить систему уравнений методом Крамера:\n",
    "а) $\\begin{cases}\n",
    "x_{1}-2x_{2}=1 \\\\\n",
    "3x_{1}-4x_{2}=7\n",
    "\\end{cases}$"
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    "**Решение 5а):**\n",
    "\n",
    "Найдем определитель матрицы:\n",
    "\n",
    "$detA=\\begin{vmatrix}\n",
    "1 & -2\\\\\n",
    "3 & -4\\\\\n",
    "\\end{vmatrix}=(1 \\cdot -4) - (-2 \\cdot 3) = 2$\n",
    "\n",
    "Найдем определитель $detA_{1}$:\n",
    "\n",
    "$detA_{1}=\\begin{vmatrix}\n",
    "1 & -2\\\\\n",
    "7 & -4\\\\\n",
    "\\end{vmatrix}=(1 \\cdot -4) - (-2 \\cdot 7) = 10$\n",
    "\n",
    "Найдем определитель $detA_{2}$:\n",
    "\n",
    "$detA_{2}=\\begin{vmatrix}\n",
    "1 & 1\\\\\n",
    "3 & 1\\\\\n",
    "\\end{vmatrix}=(1 \\cdot 7) - (1 \\cdot 3) = 4$\n",
    "\n",
    "Найдем значения $x_{1}, x_{2}$:\n",
    "\n",
    "$x_{1}=10/2=5$\n",
    "\n",
    "$x_{2}=4/2=2$\n",
    "\n",
    "Проверим полученное решение:\n",
    "\n",
    "$\\begin{cases}\n",
    "5-4=1 \\\\\n",
    "15-8=7\n",
    "\\end{cases}$"
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    "б) $\\begin{cases}\n",
    "2x_{1}-x_{2}+5x_{3}=10 \\\\\n",
    "x_{1}+x_{2}-3x_{3}=-2 \\\\\n",
    "2x_{1}+4x_{2}+x_{3}=1\n",
    "\\end{cases}$"
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    "**Решение 5б):**\n",
    "Найдем определитель матрицы:\n",
    "\n",
    "$detA=\\begin{vmatrix}\n",
    "2 & -1 & 5\\\\\n",
    "1 & -1 & -3\\\\\n",
    "2 & 4 & 1\n",
    "\\end{vmatrix}=(2 \\cdot 1 \\cdot 1) + (-1 \\cdot -3 \\cdot 2) + (5 \\cdot 1 \\cdot 4) - (5 \\cdot 1 \\cdot 2) - (2 \\cdot -3 \\cdot 4) - (-1 \\cdot 1 \\cdot 1) = 43$\n",
    "\n",
    "Найдем определитель $detA_{1}$:\n",
    "\n",
    "$detA_{1}=\\begin{vmatrix}\n",
    "10 & -1 & 5\\\\\n",
    "-2 & -1 & -3\\\\\n",
    "1 & 4 & 1\n",
    "\\end{vmatrix}=(10 \\cdot 1 \\cdot 1) + (-1 \\cdot -3 \\cdot 1) + (5 \\cdot -2 \\cdot 4) - (5 \\cdot 1 \\cdot 1) - (10 \\cdot -3 \\cdot 4) - (-1 \\cdot -2 \\cdot 1) = 86$\n",
    "\n",
    "\n",
    "Найдем определитель $detA_{2}$:\n",
    "\n",
    "$detA_{2}=\\begin{vmatrix}\n",
    "2 & 10 & 5\\\\\n",
    "1 & -2 & -3\\\\\n",
    "2 & 1 & 1\n",
    "\\end{vmatrix}=(2 \\cdot -2 \\cdot 1) + (10 \\cdot -3 \\cdot 2) + (5 \\cdot 1 \\cdot 1) - (5 \\cdot -2 \\cdot 2) - (2 \\cdot -3 \\cdot 1) - (10 \\cdot 1 \\cdot 1) = -43$\n",
    "\n",
    "\n",
    "Найдем определитель $detA_{3}$:\n",
    "\n",
    "$detA_{3}=\\begin{vmatrix}\n",
    "2 & -1 & 10\\\\\n",
    "1 & -1 & -2\\\\\n",
    "2 & 4 & 1\n",
    "\\end{vmatrix}=(2 \\cdot 1 \\cdot 1) + (-1 \\cdot -2 \\cdot 2) + (10 \\cdot 1 \\cdot 4) - (10 \\cdot 1 \\cdot 2) - (2 \\cdot -2 \\cdot 4) - (-1 \\cdot 1 \\cdot 1) = 43$\n",
    "\n",
    "Найдем значения $x_{1}, x_{2}, x_{3}$:\n",
    "\n",
    "$x_{1}=86/43=2$\n",
    "\n",
    "$x_{2}=-43/43=-1$\n",
    "\n",
    "$x_{3}=43/43=1$\n",
    "\n",
    "Проверим полученное решение:\n",
    "\n",
    "$\\begin{cases}\n",
    "4+1+5=10 \\\\\n",
    "2-1-3=-2 \\\\\n",
    "4-4+1=1\n",
    "\\end{cases}$\n",
    "\n"
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