Элементы обратной матрицы вычисляются по формуле 
Найдем определитель:
Найдем алгебраические дополнения:
![\big A_{11}=(-1)^{1+1}\cdot |2|=2
\\\
\big A_{12}=(-1)^{1+2}\cdot |5|=-5
\\\
\big A_{21}=(-1)^{2+1}\cdot |1|=-1
\\\
\big A_{22}=(-1)^{2+2}\cdot |3|=3](https://tex.z-dn.net/?f=%5Cbig+A_%7B11%7D%3D%28-1%29%5E%7B1%2B1%7D%5Ccdot+%7C2%7C%3D2%0A%5C%5C%5C%0A%5Cbig+A_%7B12%7D%3D%28-1%29%5E%7B1%2B2%7D%5Ccdot+%7C5%7C%3D-5%0A%5C%5C%5C%0A%5Cbig+A_%7B21%7D%3D%28-1%29%5E%7B2%2B1%7D%5Ccdot+%7C1%7C%3D-1%0A%5C%5C%5C%0A%5Cbig+A_%7B22%7D%3D%28-1%29%5E%7B2%2B2%7D%5Ccdot+%7C3%7C%3D3 "\big A_{11}=(-1)^{1+1}\cdot |2|=2
\\\
\big A_{12}=(-1)^{1+2}\cdot |5|=-5
\\\
\big A_{21}=(-1)^{2+1}\cdot |1|=-1
\\\
\big A_{22}=(-1)^{2+2}\cdot |3|=3")
Обратная матрица составляется как транспонированная матрица алгебраических дополнений, домноженная на величину, обратную определителю:
Определитель:
![|B|=\left|\begin{array}{ccc}4&-1&2\\1&1&-2\\0&-1&3\end{array}\right|=4\cdot1\cdot3+0\cdot(-1)\cdot(-2)+2\cdot1\cdot(-1)-
\\\
-2\cdot1\cdot0-4\cdot(-2)\cdot(-1)-3\cdot(-1)\cdot 1=
12-2-8+3=5](https://tex.z-dn.net/?f=%7CB%7C%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D4%26-1%262%5C%5C1%261%26-2%5C%5C0%26-1%263%5Cend%7Barray%7D%5Cright%7C%3D4%5Ccdot1%5Ccdot3%2B0%5Ccdot%28-1%29%5Ccdot%28-2%29%2B2%5Ccdot1%5Ccdot%28-1%29-%0A%5C%5C%5C%0A-2%5Ccdot1%5Ccdot0-4%5Ccdot%28-2%29%5Ccdot%28-1%29-3%5Ccdot%28-1%29%5Ccdot+1%3D%0A12-2-8%2B3%3D5 "|B|=\left|\begin{array}{ccc}4&-1&2\\1&1&-2\\0&-1&3\end{array}\right|=4\cdot1\cdot3+0\cdot(-1)\cdot(-2)+2\cdot1\cdot(-1)-
\\\
-2\cdot1\cdot0-4\cdot(-2)\cdot(-1)-3\cdot(-1)\cdot 1=
12-2-8+3=5")
Алгебраические дополнения:
![\big B_{11}=(-1)^{1+1}\cdot \left|\begin{array}{ccc}1&-2\\-1&3\end{array}\right|=1\cdot3-(-2)\cdot(-1)=1
\\\
\big B_{12}=(-1)^{1+2}\cdot \left|\begin{array}{ccc}1&-2\\0&3\end{array}\right|=-(1\cdot3-0\cdot(-1))=-3
\\\
\big B_{13}=(-1)^{1+3}\cdot \left|\begin{array}{ccc}1&1\\0&-1\end{array}\right|=1\cdot(-1)-1\cdot0=-1](https://tex.z-dn.net/?f=%5Cbig+B_%7B11%7D%3D%28-1%29%5E%7B1%2B1%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D1%26-2%5C%5C-1%263%5Cend%7Barray%7D%5Cright%7C%3D1%5Ccdot3-%28-2%29%5Ccdot%28-1%29%3D1%0A%5C%5C%5C%0A%5Cbig+B_%7B12%7D%3D%28-1%29%5E%7B1%2B2%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D1%26-2%5C%5C0%263%5Cend%7Barray%7D%5Cright%7C%3D-%281%5Ccdot3-0%5Ccdot%28-1%29%29%3D-3%0A%5C%5C%5C%0A%5Cbig+B_%7B13%7D%3D%28-1%29%5E%7B1%2B3%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D1%261%5C%5C0%26-1%5Cend%7Barray%7D%5Cright%7C%3D1%5Ccdot%28-1%29-1%5Ccdot0%3D-1 "\big B_{11}=(-1)^{1+1}\cdot \left|\begin{array}{ccc}1&-2\\-1&3\end{array}\right|=1\cdot3-(-2)\cdot(-1)=1
\\\
\big B_{12}=(-1)^{1+2}\cdot \left|\begin{array}{ccc}1&-2\\0&3\end{array}\right|=-(1\cdot3-0\cdot(-1))=-3
\\\
\big B_{13}=(-1)^{1+3}\cdot \left|\begin{array}{ccc}1&1\\0&-1\end{array}\right|=1\cdot(-1)-1\cdot0=-1")
![\big B_{21}=(-1)^{2+1}\cdot \left|\begin{array}{ccc}-1&2\\-1&3\end{array}\right|=-((-1)\cdot3-2\cdot(-1))=1
\\\
\big B_{22}=(-1)^{2+2}\cdot \left|\begin{array}{ccc}4&2\\0&3\end{array}\right|=4\cdot3-2\cdot0=12
\\\
\big B_{23}=(-1)^{2+3}\cdot \left|\begin{array}{ccc}4&-1\\0&-1\end{array}\right|=-(4\cdot(-1)-(-1)\cdot0)=4](https://tex.z-dn.net/?f=%5Cbig+B_%7B21%7D%3D%28-1%29%5E%7B2%2B1%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D-1%262%5C%5C-1%263%5Cend%7Barray%7D%5Cright%7C%3D-%28%28-1%29%5Ccdot3-2%5Ccdot%28-1%29%29%3D1%0A%5C%5C%5C%0A%5Cbig+B_%7B22%7D%3D%28-1%29%5E%7B2%2B2%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D4%262%5C%5C0%263%5Cend%7Barray%7D%5Cright%7C%3D4%5Ccdot3-2%5Ccdot0%3D12%0A%5C%5C%5C%0A%5Cbig+B_%7B23%7D%3D%28-1%29%5E%7B2%2B3%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D4%26-1%5C%5C0%26-1%5Cend%7Barray%7D%5Cright%7C%3D-%284%5Ccdot%28-1%29-%28-1%29%5Ccdot0%29%3D4 "\big B_{21}=(-1)^{2+1}\cdot \left|\begin{array}{ccc}-1&2\\-1&3\end{array}\right|=-((-1)\cdot3-2\cdot(-1))=1
\\\
\big B_{22}=(-1)^{2+2}\cdot \left|\begin{array}{ccc}4&2\\0&3\end{array}\right|=4\cdot3-2\cdot0=12
\\\
\big B_{23}=(-1)^{2+3}\cdot \left|\begin{array}{ccc}4&-1\\0&-1\end{array}\right|=-(4\cdot(-1)-(-1)\cdot0)=4")
![\big B_{31}=(-1)^{3+1}\cdot \left|\begin{array}{ccc}-1&2\\1&-2\end{array}\right|=(-1)\cdot(-2)-2\cdot1=0
\\\
\big B_{32}=(-1)^{3+2}\cdot \left|\begin{array}{ccc}4&2\\1&-2\end{array}\right|=-(4\cdot(-2)-2\cdot1)=10
\\\
\big B_{33}=(-1)^{3+3}\cdot \left|\begin{array}{ccc}4&-1\\1&1\end{array}\right|=4\cdot1-(-1)\cdot1=5](https://tex.z-dn.net/?f=%5Cbig+B_%7B31%7D%3D%28-1%29%5E%7B3%2B1%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D-1%262%5C%5C1%26-2%5Cend%7Barray%7D%5Cright%7C%3D%28-1%29%5Ccdot%28-2%29-2%5Ccdot1%3D0%0A%5C%5C%5C%0A%5Cbig+B_%7B32%7D%3D%28-1%29%5E%7B3%2B2%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D4%262%5C%5C1%26-2%5Cend%7Barray%7D%5Cright%7C%3D-%284%5Ccdot%28-2%29-2%5Ccdot1%29%3D10%0A%5C%5C%5C%0A%5Cbig+B_%7B33%7D%3D%28-1%29%5E%7B3%2B3%7D%5Ccdot+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D4%26-1%5C%5C1%261%5Cend%7Barray%7D%5Cright%7C%3D4%5Ccdot1-%28-1%29%5Ccdot1%3D5 "\big B_{31}=(-1)^{3+1}\cdot \left|\begin{array}{ccc}-1&2\\1&-2\end{array}\right|=(-1)\cdot(-2)-2\cdot1=0
\\\
\big B_{32}=(-1)^{3+2}\cdot \left|\begin{array}{ccc}4&2\\1&-2\end{array}\right|=-(4\cdot(-2)-2\cdot1)=10
\\\
\big B_{33}=(-1)^{3+3}\cdot \left|\begin{array}{ccc}4&-1\\1&1\end{array}\right|=4\cdot1-(-1)\cdot1=5")
Обратная матрица:
