If $A = \begin{bmatrix} -2 & 0 & 0 \\ 1 & 2 & 3 \\ 5 & 1 & -1 \end{bmatrix}$, then the value of $|A(\text{adj } A)|$ is:

$1000$

The correct answer is Option (4) → $1000$ ##

We know that, $|A(\text{adj } A)| = |A| I_n = |A|^n$, where $I_n$ is the identity matrix of order $n$.

Here, $A = \begin{bmatrix} -2 & 0 & 0 \\ 1 & 2 & 3 \\ 5 & 1 & -1 \end{bmatrix}$

$|A| = -2(-2 - 3) - 0 - 0 = 10$

$∴A.\text{adj }(A) = |A| I_3 = 10 I$

$|A.\text{adj }(A)| = 10^3 = 1000$