If \( A \) is a square matrix of order 3 such that \( A(adj\ A)=\begin{bmatrix}-2 & 0 & 0\\0 & -2 & 0\\0 & 0 & -2\end{bmatrix} \), then \( |adj\ A| \) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4
Since \( A\,(\mathrm{adj}\,A)=|A|\,I \) and the given matrix is \( -2I \), |A| = −2.
\( \displaystyle |\mathrm{adj}\,A|=|A|^{n-1} \)
\( \displaystyle =(-2)^{2}=4 \)
Why the other options are incorrect
Option (1) \( -2 \) — Incorrect. This is |A| itself, not |adj A|.
Option (2) 2 — Incorrect. The value is \( (-2)^2=4 \), not 2.
Option (3) \( -4 \) — Incorrect. \( |\mathrm{adj}\,A|=|A|^2 \) here, so it cannot be negative; it is 4.