Let A be a square matrix of order 3 such that \(A(\mathrm{adj}\,A) = \begin{bmatrix}-4 & 0 & 0\\ 0 & -4 & 0\\ 0 & 0 & -4\end{bmatrix}\). Then the value of \(|A| + |\mathrm{adj}\,A|\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 12
For every square matrix, \(A(\mathrm{adj}\,A) = |A|\,I\). Here
\( A(\mathrm{adj}\,A) = -4\,I \)
so \(|A| = -4\).
For order n = 3, \(|\mathrm{adj}\,A| = |A|^{n-1}\):
\( |\mathrm{adj}\,A| = (-4)^{2} = 16 \)
\( |A| + |\mathrm{adj}\,A| = -4 + 16 = 12 \)
Why the other options are incorrect
Option (1) \(-68\) — Incorrect. This uses \(|\mathrm{adj}\,A| = |A|^{3} = -64\); for order 3 the power is \(n - 1 = 2\).
Option (3) \(-20\) — Incorrect. This takes \(|\mathrm{adj}\,A| = -16\); the square \((-4)^2\) is positive.
Option (4) 20 — Incorrect. This takes \(|A| = 4\); the diagonal entries of \(A(\mathrm{adj}\,A)\) give \(|A| = -4\).