Question

If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] , write  \[A^{- 1}\] in terms of A.

Answer 1

\[\left| A \right| = \begin{vmatrix}2 & 3 \\ 5 & - 2\end{vmatrix} = - 19 \neq 0\]
A is a non - singular matrix . Therefore, it is invertible . 
\[\text{ Let }C_{ij}\text{ be a cofactor of }a_{ij}\text{ in A . }\]
The cofactors of element A are given by
\[ C_{11} = - 2\]
\[ C_{12} = - 5\]
\[ C_{21} = - 3\]
\[ C_{22} = 2\]
\[adj A = \begin{bmatrix}- 2 & - 5 \\ - 3 & 2\end{bmatrix}^T = \begin{bmatrix}- 2 & - 3 \\ - 5 & 2\end{bmatrix}\]
\[ \therefore A^{- 1} = \frac{1}{\left| A \right|}adj A = \begin{bmatrix}2/19 & 3/19 \\ 5/19 & - 2/19\end{bmatrix}\]
\[\Rightarrow A^{- 1} = \frac{1}{19}A\]