A is a square matrix of order 3 such that \(|3A| = 54\). Then \(|\mathrm{adj}\,A|\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(4\)
For a matrix of order 3, each of the three rows gives a factor 3, so \(|3A| = 3^3|A|\):
\(27|A| = 54\)
\(|A| = 2\)
For order n, \(|\mathrm{adj}\,A| = |A|^{n - 1}\). With \(n = 3\):
\(|\mathrm{adj}\,A| = |A|^{2} = 2^2 = 4\)
Why the other options are incorrect
Option (1) \(2\) — Incorrect. This is \(|A|\) itself, not \(|\mathrm{adj}\,A|\).
Option (3) \(8\) — Incorrect. This uses the power 3; the power for \(|\mathrm{adj}\,A|\) is \(n - 1 = 2\).
Option (4) \(324\) — Incorrect. This takes \(|3A| = 3|A|\), giving \(|A| = 18\) and \(18^2 = 324\); the factor is \(3^3 = 27\).