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Question
Write \[A^{- 1}\text{ for }A = \begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
Solution
\[\left| A \right| = \begin{vmatrix}2 & 5 \\ 1 & 3\end{vmatrix} = 1 \neq 0\]
\[\text{ Let }C_{ij}\text{ be the cofactor of }a_{ij}\text{ in A . }\]
The cofactors of element A are given by
\[ C_{11} = 3\]
\[ C_{12} = - 1\]
\[ C_{21} = - 5\]
\[ C_{22} = 2\]
\[adj A = \begin{bmatrix}3 & - 1 \\ - 5 & 2\end{bmatrix}^T = \begin{bmatrix}3 & - 5 \\ - 1 & 2\end{bmatrix}\]
\[\left| A \right| = 6 - 5 = 1\]
\[ \therefore A^{- 1} = \frac{1}{\left| A \right|}adj A = \begin{bmatrix}3 & - 5 \\ - 1 & 2\end{bmatrix}\]
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