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Question
If \[A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\] , then the value of |adj A| is _____________ .
Options
a27
a9
a6
a2
Solution
a6
\[A = \begin{bmatrix} a & 0 & 0\\0 & a & 0\\0 & 0 & a \end{bmatrix}\]
\[ \therefore \left| A \right| = \begin{bmatrix} a & 0 & 0\\0 & a & 0\\0 & 0 & a \end{bmatrix} = a^3 \neq 0\]
and
\[n = 3\]
Thus, we have
\[\left| adj A \right| = \left| A \right|^{n - 1} = \left( a^3 \right)^2 = a^6 \]
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