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Question
Find the inverse of the matrix \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
Solution
\[\left| A \right| = \begin{vmatrix}3 & - 2 \\ - 7 & 5\end{vmatrix} = 1 \neq 0\]
\[\text{ A is a non - singular matrix . Therefore, it is invertible . }\]
\[\text{ Let }C_{ij}\text{ be a cofactor of }a_{ij}\text{ in A }. \]
The cofactors of element A are given by
\[ C_{11} = 5\]
\[ C_{12} = 7\]
\[ C_{21} = 2\]
\[ C_{22} = 3\]
\[ \therefore A^{- 1} = \frac{1}{\left| A \right|} \begin{bmatrix}5 & 7 \\ 2 & 3\end{bmatrix}^T = \begin{bmatrix}5 & 2 \\ 7 & 3\end{bmatrix}\]
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