If A is a \(3\times 3\) square matrix such that \(|A|=2\), then \(|3\,\mathrm{adj}\,A|\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 108
Two standard results for a square matrix of order n are needed:
\(|kB| = k^n|B|\)
\(|\mathrm{adj}\,A| = |A|^{n-1}\)
Here n = 3. By the adjoint result:
\(|\mathrm{adj}\,A| = |A|^{2} = 2^2 = 4\)
The scalar 3 multiplies all three rows of adj A, so it comes out as \(3^3\):
\(|3\,\mathrm{adj}\,A| = 3^3\times|\mathrm{adj}\,A|\)
\(= 27\times 4 = 108\)
Why the other options are incorrect
Option (2) 216 — Incorrect. This is \(27\times 2^3\): it uses \(|A|^n\) instead of \(|A|^{n-1}\) for the adjoint.
Option (3) 12 — Incorrect. This is \(3\times 4\): the scalar 3 is taken out only once instead of as \(3^3\).
Option (4) 6 — Incorrect. This is \(3\times|A|\): it ignores both the power of 3 and the adjoint.