![\begin{cases}2x+3y+z=1\\\ x+y-4z=0\\4x+5y-3z=1\end{cases}\\\\A \left[\begin{array}{ccc}2&3&1\\1&1&-4\\4&5&-3\end{array}\right]\ ;B \left[\begin{array}{c}1\\0\\1\end{array}\right]\ ;X \left[\begin{array}{c}x\\y\\z\end{array}\right]](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D2x%2B3y%2Bz%3D1%5C%5C%5C+x%2By-4z%3D0%5C%5C4x%2B5y-3z%3D1%5Cend%7Bcases%7D%5C%5C%5C%5CA+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%261%5C%5C1%261%26-4%5C%5C4%265%26-3%5Cend%7Barray%7D%5Cright%5D%5C+%3BB+%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D1%5C%5C0%5C%5C1%5Cend%7Barray%7D%5Cright%5D%5C+%3BX+%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx%5C%5Cy%5C%5Cz%5Cend%7Barray%7D%5Cright%5D+ "\begin{cases}2x+3y+z=1\\\ x+y-4z=0\\4x+5y-3z=1\end{cases}\\\\A \left[\begin{array}{ccc}2&3&1\\1&1&-4\\4&5&-3\end{array}\right]\ ;B \left[\begin{array}{c}1\\0\\1\end{array}\right]\ ;X \left[\begin{array}{c}x\\y\\z\end{array}\right]")
Проверяем определитель левой части: равен ли он нулю:
![\left[\begin{array}{ccc}2&3&1\\1&1&-4\\4&5&-3\end{array}\right]=-6-48+5-4+9+40\neq0](https://tex.z-dn.net/?f=+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%261%5C%5C1%261%26-4%5C%5C4%265%26-3%5Cend%7Barray%7D%5Cright%5D%3D-6-48%2B5-4%2B9%2B40%5Cneq0 "\left[\begin{array}{ccc}2&3&1\\1&1&-4\\4&5&-3\end{array}\right]=-6-48+5-4+9+40\neq0")
Метод обратной матрицы:
![AX=B|*A^{-1}\\X=A^{-1}B\\A_{11}=17\ \ \ \ \ \ A_{12}=-13\ \ \ \ \ \ A_{13}=1\\A_{21}=14\ \ \ \ \ \ A_{22}=-10\ \ \ \ \ \ A_{23}=2\\A_{31}=-13\ \ \ \ A_{32}=9\ \ \ \ \ \ \ \ \ A_{33}=-1\\\\A^T= \left[\begin{array}{ccc}17&14&-13\\-13&-10&9\\1&2&-1\end{array}\right]\\\\A^{-1}=-\frac{1}{4}*\left[\begin{array}{ccc}17&14&-13\\-13&-10&9\\1&2&-1\end{array}\right]](https://tex.z-dn.net/?f=AX%3DB%7C%2AA%5E%7B-1%7D%5C%5CX%3DA%5E%7B-1%7DB%5C%5CA_%7B11%7D%3D17%5C+%5C+%5C+%5C+%5C+%5C+A_%7B12%7D%3D-13%5C+%5C+%5C+%5C+%5C+%5C+A_%7B13%7D%3D1%5C%5CA_%7B21%7D%3D14%5C+%5C+%5C+%5C+%5C+%5C+A_%7B22%7D%3D-10%5C+%5C+%5C+%5C+%5C+%5C+A_%7B23%7D%3D2%5C%5CA_%7B31%7D%3D-13%5C+%5C+%5C+%5C+A_%7B32%7D%3D9%5C+%5C+%5C+%5C+%5C+%5C+%5C+%5C+%5C+A_%7B33%7D%3D-1%5C%5C%5C%5CA%5ET%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D17%2614%26-13%5C%5C-13%26-10%269%5C%5C1%262%26-1%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5CA%5E%7B-1%7D%3D-%5Cfrac%7B1%7D%7B4%7D%2A%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D17%2614%26-13%5C%5C-13%26-10%269%5C%5C1%262%26-1%5Cend%7Barray%7D%5Cright%5D "AX=B|*A^{-1}\\X=A^{-1}B\\A_{11}=17\ \ \ \ \ \ A_{12}=-13\ \ \ \ \ \ A_{13}=1\\A_{21}=14\ \ \ \ \ \ A_{22}=-10\ \ \ \ \ \ A_{23}=2\\A_{31}=-13\ \ \ \ A_{32}=9\ \ \ \ \ \ \ \ \ A_{33}=-1\\\\A^T= \left[\begin{array}{ccc}17&14&-13\\-13&-10&9\\1&2&-1\end{array}\right]\\\\A^{-1}=-\frac{1}{4}*\left[\begin{array}{ccc}17&14&-13\\-13&-10&9\\1&2&-1\end{array}\right]")
![X=-\frac{1}{4}*\left[\begin{array}{ccc}17&14&-13\\-13&-10&9\\1&2&-1\end{array}\right]* \left[\begin{array}{c}1\\0\\1\end{array}\right]=\left[\begin{array}{c}-1\\1\\0\end{array}\right]](https://tex.z-dn.net/?f=X%3D-%5Cfrac%7B1%7D%7B4%7D%2A%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D17%2614%26-13%5C%5C-13%26-10%269%5C%5C1%262%26-1%5Cend%7Barray%7D%5Cright%5D%2A++%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D1%5C%5C0%5C%5C1%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-1%5C%5C1%5C%5C0%5Cend%7Barray%7D%5Cright%5D "X=-\frac{1}{4}*\left[\begin{array}{ccc}17&14&-13\\-13&-10&9\\1&2&-1\end{array}\right]* \left[\begin{array}{c}1\\0\\1\end{array}\right]=\left[\begin{array}{c}-1\\1\\0\end{array}\right]")
Правило Крамера.
Находим определитель:-4
Далее находим дополнительные определители.
![a_x= \left[\begin{array}{ccc}1&3&1\\0&1&-4\\1&5&-3\end{array}\right] =-3-12-1+20=4\\\\a_y= \left[\begin{array}{ccc}2&1&1\\1&0&-4\\4&1&-3\end{array}\right] =-16+1+3+8=-4\\\\a_z= \left[\begin{array}{ccc}2&3&1\\1&1&0\\4&5&1\end{array}\right] =2+5-4-3=0\\\\X=\frac{4}{-4}=-1\ ;Y=\frac{-4}{-4}=1\ ;Z=\frac{0}{-4}=0\\OTBET: \left[\begin{array}{ccc}-1\\1\\0\end{array}\right]](https://tex.z-dn.net/?f=a_x%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%263%261%5C%5C0%261%26-4%5C%5C1%265%26-3%5Cend%7Barray%7D%5Cright%5D+%3D-3-12-1%2B20%3D4%5C%5C%5C%5Ca_y%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%261%261%5C%5C1%260%26-4%5C%5C4%261%26-3%5Cend%7Barray%7D%5Cright%5D+%3D-16%2B1%2B3%2B8%3D-4%5C%5C%5C%5Ca_z%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%261%5C%5C1%261%260%5C%5C4%265%261%5Cend%7Barray%7D%5Cright%5D+%3D2%2B5-4-3%3D0%5C%5C%5C%5CX%3D%5Cfrac%7B4%7D%7B-4%7D%3D-1%5C+%3BY%3D%5Cfrac%7B-4%7D%7B-4%7D%3D1%5C+%3BZ%3D%5Cfrac%7B0%7D%7B-4%7D%3D0%5C%5COTBET%3A+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-1%5C%5C1%5C%5C0%5Cend%7Barray%7D%5Cright%5D+ "a_x= \left[\begin{array}{ccc}1&3&1\\0&1&-4\\1&5&-3\end{array}\right] =-3-12-1+20=4\\\\a_y= \left[\begin{array}{ccc}2&1&1\\1&0&-4\\4&1&-3\end{array}\right] =-16+1+3+8=-4\\\\a_z= \left[\begin{array}{ccc}2&3&1\\1&1&0\\4&5&1\end{array}\right] =2+5-4-3=0\\\\X=\frac{4}{-4}=-1\ ;Y=\frac{-4}{-4}=1\ ;Z=\frac{0}{-4}=0\\OTBET: \left[\begin{array}{ccc}-1\\1\\0\end{array}\right]")
Метод Гаусса:
Записываем систему как расширенную матрицу и изменяем ее путем элементарных преобразований к единичной в левой части:
\left[\begin{array}{ccccc}1&1&-4&|&0\\2&3&1&|&1\\4&5&-3&|&1\end{array}\right]=>\left[\begin{array}{ccccc}1&1&-4&|&0\\0&1&9&|&1\\0&1&13&|&1\end{array}\right]=>\\\\\left[\begin{array}{ccccc}1&0&-13&|&-1\\0&1&9&|&1\\0&0&4&|&0\end{array}\right]" alt="\left[\begin{array}{ccccc}2&3&1&|&1\\1&1&-4&|&0\\4&5&-3&|&1\end{array}\right]=>\left[\begin{array}{ccccc}1&1&-4&|&0\\2&3&1&|&1\\4&5&-3&|&1\end{array}\right]=>\left[\begin{array}{ccccc}1&1&-4&|&0\\0&1&9&|&1\\0&1&13&|&1\end{array}\right]=>\\\\\left[\begin{array}{ccccc}1&0&-13&|&-1\\0&1&9&|&1\\0&0&4&|&0\end{array}\right]" align="absmiddle" class="latex-formula">