Let A be a square matrix of order 3 such that \(|A| = 3\). Then the value of \(|\mathrm{adj}(2A)|\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 576
Apply \(|\mathrm{adj}\,M| = |M|^{n-1}\) to the matrix M = 2A, with n = 3.
First, by \(|kA| = k^{n}|A|\):
\( |2A| = 2^{3} \times 3 = 24 \)
Then
\( |\mathrm{adj}(2A)| = |2A|^{2} \)
\( = 24^{2} = 576 \)
Why the other options are incorrect
Option (1) 36 — Incorrect. This takes \(|2A| = 2 \times 3 = 6\); the factor 2 must be cubed for order 3.
Option (2) 144 — Incorrect. This takes \(|2A| = 2^{2} \times 3 = 12\), the rule for order 2.
Option (4) 72 — Incorrect. This is \(2^{3} \times |A|^{2}\); the factor \(2^{3}\) must also be squared.