If $A=\begin{bmatrix}a&a&a\\0&a&a\\0&0&a\end{bmatrix}$, then $|adj\, A|$ is equal to
Answer & explanation
Correct answer: option 2
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, $|\text{adj }A| = a^{6}$