294 lines
8.4 KiB
Markdown
294 lines
8.4 KiB
Markdown
## 20 номер – П 2.2.8
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#семестр_1 #высшая_математика
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### Пример:
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МЕТОД ОБРАТНОЙ МАТРИЦЫ
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$$
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\begin{cases}
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x+2y+3z=5 \\
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4x+5y+6z=8 \\
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7x+8y=2
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0\end{pmatrix}$
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$x=\begin{pmatrix}x \\ y \\ z\end{pmatrix}$
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$b=\begin{pmatrix}5 \\ 8 \\ 2\end{pmatrix}$
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$\det A=27\neq 0$
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$A^{-1}=\begin{pmatrix}-1 \dfrac{7}{9} & \dfrac{8}{9} & -\dfrac{1}{9} \\ 1 \dfrac{5}{9} & -\dfrac{7}{9} & \dfrac{2}{9} \\ -\dfrac{1}{9} & \dfrac{2}{9} & -\dfrac{1}{9}\end{pmatrix}$
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$x=A^{-1}b=\begin{pmatrix}-1 \dfrac{7}{9} & \dfrac{8}{9} & -\dfrac{1}{9} \\ 1 \dfrac{5}{9} & -\dfrac{7}{9} & \dfrac{2}{9} \\ -\dfrac{1}{9} & \dfrac{2}{9} & -\dfrac{1}{9}\end{pmatrix}\cdot \begin{pmatrix}5 \\ 8 \\ 2\end{pmatrix}=\begin{pmatrix}-2 \\ 2 \\ 1\end{pmatrix}$
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$\text{Ответ: }x=-2; \ y=2; \ z=1;$
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## 21 номер – П 2.2.9
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### Пример:
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$$
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\begin{cases}
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2x_{1}-3x_{2}+x_{3}=-7 \\
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x_{1}+2x_{2}-3x_{3}=14 \\
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-x_{1}-x_{2}+5x_{3}=-18
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}2 & -3 & 1 \\ 1 & 2 & -3 \\ -1 & -1 & 5\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3}\end{pmatrix}$
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$b=\begin{pmatrix}-7 \\ 14 \\ -18\end{pmatrix}$
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$\det A=21\neq 0$
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$A^{-1}=\begin{pmatrix}\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{1}{3} \\ -\dfrac{2}{21} & \dfrac{11}{21} & \dfrac{1}{3} \\ \dfrac{1}{21} & \dfrac{5}{21} & \dfrac{1}{3}\end{pmatrix}$
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$x=A^{-1}b=\begin{pmatrix}\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{1}{3} \\ -\dfrac{2}{21} & \dfrac{11}{21} & \dfrac{1}{3} \\ \dfrac{1}{21} & \dfrac{5}{21} & \dfrac{1}{3}\end{pmatrix}\cdot\begin{pmatrix}-7 \\ 14 \\ -18\end{pmatrix}=\begin{pmatrix}1 \\ 2 \\ -3\end{pmatrix}$
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$\text{Ответ: }x_{1}=1;\ x_{2}=2;\ x_{3}=-3;$
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## 22 номер – П 2.2.9
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### Пример:
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$$
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\begin{cases}
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2x_{1}-3x_{2}+x_{3}=-7 \\
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x_{1}+2x_{2}-3x_{3}=14 \\
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-x_{1}-x_{2}+5x_{3}=-18
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}2 & -3 & 1 \\ 1 & 2 & -3 \\ -1 & -1 & 5\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3}\end{pmatrix}$
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$b=\begin{pmatrix}-7 \\ 14 \\ -18\end{pmatrix}$
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$\det A=21\neq 0$
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$A^{-1}=\begin{pmatrix}\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{1}{3} \\ -\dfrac{2}{21} & \dfrac{11}{21} & \dfrac{1}{3} \\ \dfrac{1}{21} & \dfrac{5}{21} & \dfrac{1}{3}\end{pmatrix}$
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$x=A^{-1}b=\begin{pmatrix}\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{1}{3} \\ -\dfrac{2}{21} & \dfrac{11}{21} & \dfrac{1}{3} \\ \dfrac{1}{21} & \dfrac{5}{21} & \dfrac{1}{3}\end{pmatrix}\cdot\begin{pmatrix}-7 \\ 14 \\ -18\end{pmatrix}=\begin{pmatrix}1 \\ 2 \\ -3\end{pmatrix}$
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$\text{Ответ: }x_{1}=1;\ x_{2}=2;\ x_{3}=-3;$
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:D
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## 23 номер – П 2.2.10
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### Пример:
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$$
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\begin{cases}
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2x_{1}+x_{2}-x_{3}=3 \\
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x_{1}+3x_{2}+2x_{3}=-1 \\
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x_{1}+x_{2}=5
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}2 & 1 & -1 \\ 1 & 3 & 2 \\ 1 & 1 & 0\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3}\end{pmatrix}$
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$b=\begin{pmatrix}3 \\ -1 \\ 5\end{pmatrix}$
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$\det A=2(3\cdot0-2\cdot1)-1(1\cdot0-2\cdot1)+(-1)(1\cdot1-3\cdot1)=0$
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$\det A=0; A^{-1}\ \text{не существует}$
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$\text{Ответ: решений нет}$
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## 24 номер – П 2.2.11
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### Пример:
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$$
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\begin{cases}
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x_{1}+2x_{2}+3x_{3}=3 \\
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2x_{1}+6x_{1}+4x_{3}=6 \\
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3x_{1}+10x_{2}+8x_{3}=21
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}1 & 2 & 3 \\ 8 & 0 & 4 \\ 3 & 10 & 8\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3}\end{pmatrix}$
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$b=\begin{pmatrix}3 \\ 6 \\ 21\end{pmatrix}$
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$\det A=96\neq 0$
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$A^{-1}=\begin{pmatrix}-\dfrac{5}{12} & \dfrac{7}{48} & \dfrac{1}{12} \\ -\dfrac{13}{24} & -\dfrac{1}{96} & \dfrac{5}{24} \\ \dfrac{5}{6} & -\dfrac{1}{24} & -\dfrac{1}{6}\end{pmatrix}$
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$x=A^{-1}b=\begin{pmatrix}-\dfrac{5}{12} & \dfrac{7}{48} & \dfrac{1}{12} \\ -\dfrac{13}{24} & -\dfrac{1}{96} & \dfrac{5}{24} \\ \dfrac{5}{6} & -\dfrac{1}{24} & -\dfrac{1}{6}\end{pmatrix}\cdot\begin{pmatrix}3 \\ 6 \\ 21\end{pmatrix}=\begin{pmatrix}1\dfrac{3}{8} \\ 2\dfrac{11}{16} \\ -1\dfrac{1}{4}\end{pmatrix}$
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$\text{Ответ: }x_{1}=1\dfrac{3}{8}; \ x_{2}=2\dfrac{11}{16}; \ x_{3}=-1\dfrac{1}{4};$
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## 25 номер – П 2.2.12
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### Пример:
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$$
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\begin{cases}
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ax_{1}+x_{2}+x_{3}=1 \\
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x_{1}+ax_{2}+x_{3}=a \\
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x_{1}+x_{2}+ax_{3}=a^{2}
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3}\end{pmatrix}$
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$b=\begin{pmatrix}1 \\ a \\ a^{2}\end{pmatrix}$
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$R_2=R_2-R_1,\ R_3=R_3-R_1$
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$\det A=\det\begin{pmatrix}a & 1 & 1 \\ 1-a & a-1 & 0 \\ 1-a & 0 & a-1\end{pmatrix}$
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$\det A=(a-1)^{2}\det\begin{pmatrix}a & 1 & 1 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{pmatrix}=(a-1)^{2}(a+2)$
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$a\neq1,\ a\neq-2\ ; \ \det A\neq0$
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$A^{-1}=\dfrac{1}{(a-1)(a+2)}\begin{pmatrix}a+1 & -1 & -1 \\ -1 & a+1 & -1 \\ -1 & -1 & a+1\end{pmatrix}$
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$x=A^{-1}b=\dfrac{1}{(a-1)(a+2)}\begin{pmatrix}a+1 & -1 & -1 \\ -1 & a+1 & -1 \\ -1 & -1 &a+1\end{pmatrix}\cdot\begin{pmatrix}1 \\ a \\ a^{2}\end{pmatrix}=\dfrac{1}{(a-1)(a+2)}\begin{pmatrix}1-a^{2} \\ a-1 \\ (a-1)(a+1)^{2}\end{pmatrix}=\begin{pmatrix}-\dfrac{a+1}{a+2} \\ \dfrac{1}{a+2} \\ \dfrac{(a+1)^{2}}{a+2}\end{pmatrix}$
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$a=1; \det A=0$
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$\begin{cases}x_{1}+x_{2}+x_{3}=1\\x_{1}+x_{2}+x_{3}=1\\x_{1}+x_{2}+x_{3}=1\end{cases}$
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$x_{1}=1-x_{2}-x_{3}$
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$a=-2; \det A=0$
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$\begin{cases}-2x_{1}+x_{2}+x_{3}=1\\x_{1}-2x_{2}+x_{3}=-2\\x_{1}+x_{2}-2x_{3}=4\end{cases}$
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$\text{Решений нет}$
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## 26 номер – П 2.2.13
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### Пример:
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$$
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\begin{cases}
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3x_{1}-5x_{2}+2x_{3}-4x_{4}=0 \\
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-3x_{1}+4x_{2}-5x_{3}+3x_{4}=-2 \\
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-5x_{1}+7x_{2}-7x_{3}+5x_{4}=-2 \\
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8x_{1}-8x_{2}+5x_{3}-6x_{4}=-5
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}3 & -5 & 2 & -4 \\ -3 & 4 & -5 & 3 \\ -5 & 7 & -7 & 5 \\ 8 & -8 & 5 & -6\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{pmatrix}$
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$b=\begin{pmatrix}0 \\ -2 \\ -2 \\ -5\end{pmatrix}$
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$\det A=17\neq 0$
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$A^{-1}=\dfrac{1}{17}\begin{pmatrix}-5 & -3 & 5 & 6 \\ 8 & -53 & 43 & 4 \\ -2 & -8 & 2 & -1 \\ -19 & 60 & -49 & -1\end{pmatrix}$
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$x=A^{-1}b=\dfrac{1}{17}\begin{pmatrix}-5 & -3 & 5 & 6 \\ 8 & -53 & 43 & 4 \\ -2 & -8 & 2 & -1 \\ -19 & 60 & -49 & -1\end{pmatrix}\cdot\begin{pmatrix}0 \\ -2 \\ -2 \\ -5\end{pmatrix}=\begin{pmatrix}-2 \\ 0 \\ 1 \\ -1\end{pmatrix}$
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$\text{Ответ: }x_{1}=-2; \ x_{2}=0; \ x_{3}=1; \ x_{4}=-1;$
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## 27 номер – П 2.2.14
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### Пример:
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$$
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\begin{cases}
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6x_{1}-5x_{2}+4x_{3}+7x_{4}=28 \\
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5x_{1}-8x_{2}+5x_{3}+8x_{4}=36 \\
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9x_{1}-8x_{2}+5x_{3}+10x_{4}=42 \\
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3x_{1}+2x_{2}+2x_{3}+2x_{4}=2
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}6 & -5 & 4 & 7 \\ 5 & -8 & 5 & 8 \\ 9 & -8 & 5 & 10 \\ 3 & 2 & 2 & 2\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{pmatrix}$
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$b=\begin{pmatrix}28 \\ 36 \\ 42 \\ 2\end{pmatrix}$
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$\det A=-6\neq 0$
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$A^{-1}=\dfrac{1}{6}\begin{pmatrix}-52 & 12 & 24 & 14 \\ 34 & -9 & -15 & -8 \\ -60 & 18 & 24 & 18 \\ 104 & -27 & -45 & -28\end{pmatrix}$
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$x=A^{-1}b=\dfrac{1}{6}\begin{pmatrix}-52 & 12 & 24 & 14 \\ 34 & -9 & -15 & -8 \\ -60 & 18 & 24 & 18 \\ 104 & -27 & -45 & -28\end{pmatrix}\cdot\begin{pmatrix}28 \\ 36 \\ 42 \\ 2\end{pmatrix}=\begin{pmatrix}2 \\ -3 \\ 2 \\ -1\end{pmatrix}$
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$\text{Ответ: }x_{1}=2; \ x_{2}=-3; \ x_{3}=2; \ x_{4}=-1;$
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## 28 номер – П 2.2.15
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### Пример:
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$$
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\begin{cases}
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2x_{1}+6x_{2}+x_{3}=0 \\
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x_{1}+2x_{2}-2x_{3}+4x_{4}=0 \\
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-x_{1}+4x_{2}+5x_{3}-4x_{4}=0 \\
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3x_{1}+x_{3}+2x_{4}=0
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}2 & 6 & 1 & 0 \\ 1 & 2 & -2 & 4 \\ -1 & 4 & 5 & -4 \\ 3 & 0 & 1 & 2\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{pmatrix}$
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$b=\begin{pmatrix}0 \\ 0 \\ 0 \\ 0\end{pmatrix}$
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$\det A=240\neq 0$
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$A^{-1}\ \text{существует}$
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$x=A^{-1}b=A^{-1}\cdot\begin{pmatrix}0 \\ 0 \\ 0 \\ 0\end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0 \\ 0\end{pmatrix}$
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$\text{Ответ: }x_{1}=0; \ x_{2}=0; \ x_{3}=0; \ x_{4}=0;$
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## 29 номер – П 2.2.27
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### Пример:
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$$
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\begin{cases}
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2x_{1}+x_{2}+4x_{3}+8x_{4}=0 \\
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x_{1}+3x_{2}-6x_{3}+2x_{4}=0 \\
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3x_{1}-2x_{2}+2x_{3}-2x_{4}=0 \\
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2x_{1}-x_{2}+2x_{3}=0
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\end{cases}
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$$
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### Решение:
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$A=\begin{pmatrix}2 & 1 & 4 & 8 \\ 1 & 3 & -6 & 2 \\ 3 & -2 & 2 & -2 \\ 2 & -1 & 2 & 0\end{pmatrix}$
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$x=\begin{pmatrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{pmatrix}$
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$b=\begin{pmatrix}0 \\ 0 \\ 0 \\ 0\end{pmatrix}$
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$\det A=24\neq 0$
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$A^{-1}\ \text{существует}$
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$x=A^{-1}b=A^{-1}\cdot\begin{pmatrix}0 \\ 0 \\ 0 \\ 0\end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0 \\ 0\end{pmatrix}$
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$\text{Ответ: }x_{1}=0;\ x_{2}=0;\ x_{3}=0;\ x_{4}=0;$
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