If $A=\begin{bmatrix}a&a&a\\0&a&a\\0&0&a\end{bmatrix}$, then $|adj\, A|$ is equal to

$a^6$

The correct answer is Option (2) → $a^6$

Given:

$A = \begin{bmatrix} a & a & a \\ 0 & a & a \\ 0 & 0 & a \end{bmatrix}$

$A$ is an upper triangular matrix.

For an upper triangular matrix, $\det(A)$ is the product of diagonal elements:

$|A| = a \cdot a \cdot a = a^{3}$

For a $3\times3$ matrix, $|\text{adj }A| = |A|^{2}$

$\Rightarrow |\text{adj }A| = (a^{3})^{2} = a^{6}$

Hence, $|\text{adj }A| = a^{6}$