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Question
If \[A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}\], write adj A.
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Solution
\[\left| A \right| = \begin{vmatrix}1 & - 3 \\ 2 & 0\end{vmatrix} = 6 \neq 0\]
\[\text{ A is a non - singular matrix . Therefore, it is invertible . }\]
\[\text{ Let }C_{ij}\text{ be a cofactor of }a_{ij}\text{ in A .} \]
The cofactors of element A are given by
\[ C_{11} = 0\]
\[ C_{12} = - 2\]
\[ C_{21} = 3\]
\[ C_{22} = 1\]
\[ \therefore adj A = \begin{bmatrix}0 & - 2 \\ 3 & 1\end{bmatrix}^T = \begin{bmatrix}0 & 3 \\ - 2 & 1\end{bmatrix}\]
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