Обратную матрицу найдем по формуле:
,
где |A| - определитель матрицы, а 
- транспонированная матрица алгебраических дополнений
![|A|=\left[\begin{array}{ccc}2&3&-1\\1&-1&3\\3&5&1\end{array}\right]=-2+27-5-3-30-3=-16](https://tex.z-dn.net/?f=%7CA%7C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%26-1%5C%5C1%26-1%263%5C%5C3%265%261%5Cend%7Barray%7D%5Cright%5D%3D-2%2B27-5-3-30-3%3D-16 "|A|=\left[\begin{array}{ccc}2&3&-1\\1&-1&3\\3&5&1\end{array}\right]=-2+27-5-3-30-3=-16")
Т.к. определитель матрицы не равен 0, то обратная матрица существует.
Находим матрицу миноров. Для каждого элемента матрицы соответствующий ему минор вычисляется по определителю матрицы 2х2, которая получается вычеркиванием соответствующей строки и столбца для этого элемента:
![m_{11}=\left[\begin{array}{cc}-1&3\\5&1\end{array}\right]=-1-15=-16\\m_{12}=\left[\begin{array}{cc}1&3\\3&1\end{array}\right]=1-9=-8\\m_{13}=\left[\begin{array}{cc}1&-1\\3&5\end{array}\right]=5+3=8](https://tex.z-dn.net/?f=m_%7B11%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-1%263%5C%5C5%261%5Cend%7Barray%7D%5Cright%5D%3D-1-15%3D-16%5C%5Cm_%7B12%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%263%5C%5C3%261%5Cend%7Barray%7D%5Cright%5D%3D1-9%3D-8%5C%5Cm_%7B13%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26-1%5C%5C3%265%5Cend%7Barray%7D%5Cright%5D%3D5%2B3%3D8 "m_{11}=\left[\begin{array}{cc}-1&3\\5&1\end{array}\right]=-1-15=-16\\m_{12}=\left[\begin{array}{cc}1&3\\3&1\end{array}\right]=1-9=-8\\m_{13}=\left[\begin{array}{cc}1&-1\\3&5\end{array}\right]=5+3=8")
![m_{21}=\left[\begin{array}{cc}3&-1\\5&1\end{array}\right]=3+5=8\\m_{22}=\left[\begin{array}{cc}2&-1\\3&1\end{array}\right]=2+3=5\\m_{23}=\left[\begin{array}{cc}2&3\\3&5\end{array}\right]=10-9=1](https://tex.z-dn.net/?f=m_%7B21%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%26-1%5C%5C5%261%5Cend%7Barray%7D%5Cright%5D%3D3%2B5%3D8%5C%5Cm_%7B22%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%26-1%5C%5C3%261%5Cend%7Barray%7D%5Cright%5D%3D2%2B3%3D5%5C%5Cm_%7B23%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%263%5C%5C3%265%5Cend%7Barray%7D%5Cright%5D%3D10-9%3D1 "m_{21}=\left[\begin{array}{cc}3&-1\\5&1\end{array}\right]=3+5=8\\m_{22}=\left[\begin{array}{cc}2&-1\\3&1\end{array}\right]=2+3=5\\m_{23}=\left[\begin{array}{cc}2&3\\3&5\end{array}\right]=10-9=1")
![m_{31}=\left[\begin{array}{cc}3&-1\\-1&3\end{array}\right]=9-1=8\\m_{32}=\left[\begin{array}{cc}2&-1\\1&3\end{array}\right]=6+1=7\\m_{33}=\left[\begin{array}{cc}2&3\\1&-1\end{array}\right]=-2-3=-5](https://tex.z-dn.net/?f=m_%7B31%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%26-1%5C%5C-1%263%5Cend%7Barray%7D%5Cright%5D%3D9-1%3D8%5C%5Cm_%7B32%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%26-1%5C%5C1%263%5Cend%7Barray%7D%5Cright%5D%3D6%2B1%3D7%5C%5Cm_%7B33%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D2%263%5C%5C1%26-1%5Cend%7Barray%7D%5Cright%5D%3D-2-3%3D-5 "m_{31}=\left[\begin{array}{cc}3&-1\\-1&3\end{array}\right]=9-1=8\\m_{32}=\left[\begin{array}{cc}2&-1\\1&3\end{array}\right]=6+1=7\\m_{33}=\left[\begin{array}{cc}2&3\\1&-1\end{array}\right]=-2-3=-5")
Получили следующую матрицу миноров:
![M=\left[\begin{array}{ccc}-16&-8&8\\8&5&1\\8&7&-5\end{array}\right]](https://tex.z-dn.net/?f=M%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-16%26-8%268%5C%5C8%265%261%5C%5C8%267%26-5%5Cend%7Barray%7D%5Cright%5D "M=\left[\begin{array}{ccc}-16&-8&8\\8&5&1\\8&7&-5\end{array}\right]")
Из матрицы миноров получим матрицу алгебраических дополнений заменой знака на противоположный у элементов матрицы миноров, у которых сумма номеров строк и столбца нечетна:
![\tilde{A}=\left[\begin{array}{ccc}-16&8&8\\-8&5&-1\\8&-7&-5\end{array}\right]](https://tex.z-dn.net/?f=%5Ctilde%7BA%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-16%268%268%5C%5C-8%265%26-1%5C%5C8%26-7%26-5%5Cend%7Barray%7D%5Cright%5D "\tilde{A}=\left[\begin{array}{ccc}-16&8&8\\-8&5&-1\\8&-7&-5\end{array}\right]")
Следующим шагом получаем транспонированную матрицу алгебраических дополнений:
![\tilde{A^T}=\left[\begin{array}{ccc}-16&-8&8\\8&5&-7\\8&-1&-5\end{array}\right]](https://tex.z-dn.net/?f=%5Ctilde%7BA%5ET%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-16%26-8%268%5C%5C8%265%26-7%5C%5C8%26-1%26-5%5Cend%7Barray%7D%5Cright%5D "\tilde{A^T}=\left[\begin{array}{ccc}-16&-8&8\\8&5&-7\\8&-1&-5\end{array}\right]")
Обратная матрица:
![A^{-1}=-\frac{1}{16}\left[\begin{array}{ccc}-16&-8&8\\8&5&-7\\8&-1&-5\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%3D-%5Cfrac%7B1%7D%7B16%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-16%26-8%268%5C%5C8%265%26-7%5C%5C8%26-1%26-5%5Cend%7Barray%7D%5Cright%5D "A^{-1}=-\frac{1}{16}\left[\begin{array}{ccc}-16&-8&8\\8&5&-7\\8&-1&-5\end{array}\right]")
Проверим, что произведение исходной и обратной матрицы равно единичной:
![A*A^{-1}=-\frac{1}{16}\left[\begin{array}{ccc}2&3&-1\\1&-1&3\\3&5&1\end{array}\right]\left[\begin{array}{ccc}-16&-8&8\\8&5&-7\\8&-1&-5\end{array}\right]=-\frac{1}{16}*\left[\begin{array}{ccc}-16&0&0\\0&-16&0\\0&0&-16\end{array}\right]=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]](https://tex.z-dn.net/?f=A%2AA%5E%7B-1%7D%3D-%5Cfrac%7B1%7D%7B16%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%26-1%5C%5C1%26-1%263%5C%5C3%265%261%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-16%26-8%268%5C%5C8%265%26-7%5C%5C8%26-1%26-5%5Cend%7Barray%7D%5Cright%5D%3D-%5Cfrac%7B1%7D%7B16%7D%2A%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-16%260%260%5C%5C0%26-16%260%5C%5C0%260%26-16%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%260%5C%5C0%261%260%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%5D "A*A^{-1}=-\frac{1}{16}\left[\begin{array}{ccc}2&3&-1\\1&-1&3\\3&5&1\end{array}\right]\left[\begin{array}{ccc}-16&-8&8\\8&5&-7\\8&-1&-5\end{array}\right]=-\frac{1}{16}*\left[\begin{array}{ccc}-16&0&0\\0&-16&0\\0&0&-16\end{array}\right]=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]")