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

Answer 1

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

\[\left| adj A \right| = \left| A \right|^{n - 1} = \left( a^3 \right)^2 = a^6 \]