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Question
If A is a square matrix such that \[A \left( adj A \right) = \begin{bmatrix}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{bmatrix}\] , then write the value of |adj A|.
Solution
\[\text{ Given: }\hspace{0.167em} A\left( adj A \right) = \begin{bmatrix} 5 & 0 & 0\\0 & 5 & 0\\0 & 0 & 5 \end{bmatrix} \]
\[ \Rightarrow \hspace{0.167em} \left| A \right| I_n = 5\begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{bmatrix}\]
\[ \Rightarrow \hspace{0.167em} \left| A \right| = 5\]
\[\text{Now,}\left| adj A \right| = \hspace{0.167em} \left| A \right|^{n - 1} = 5^{3 - 1} = 25 \]
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