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Question
For any 2 × 2 matrix, if \[A \left( adj A \right) = \begin{bmatrix}10 & 0 \\ 0 & 10\end{bmatrix}\] , then |A| is equal to ______ .
Options
20
100
10
0
Solution
10
\[A\left( adj A \right) = \begin{bmatrix} 10 & 0\\0 & 10 \end{bmatrix}\]
By definition, we have
\[A\left( adj A \right) = \left| A \right|I = \left( adj A \right)A (\text{ Where I is the identity matrix })\]
\[ \Rightarrow \left| A \right|I = A\left( adj A \right)\]
\[ \Rightarrow \left| A \right|I = 10\begin{bmatrix} 1 & 0\\0 & 1 \end{bmatrix}\]
\[ \Rightarrow \left| A \right| = 10\]
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