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Question
If \[A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}, B = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] , find adj (AB).
Solution
\[A \times B = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\]
\[A \times B\text{ is a non - singular matrix . Therefore, it is invertible }. \]
\[\text{ Let }C_{ij}\text{ be a cofactor of }a_{ij}\text{ in A . }\]
The cofactors of element A are given by
\[ C_{11} = d\]
\[ C_{12} = - c\]
\[ C_{21} = - b\]
\[ C_{22} = a\]
\[ \therefore adj A = \begin{bmatrix}d & - c \\ - b & a\end{bmatrix}^T = \begin{bmatrix}d & - b \\ - c & a\end{bmatrix}\]
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