If A is a square matrix of order 3 with \(|A| = 2\), then \(|\mathrm{adj}(\mathrm{adj}\,A)|\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 16
For a matrix M of order n, \(|\mathrm{adj}\,M| = |M|^{\,n - 1}\). It is used twice, with n = 3.
\(|\mathrm{adj}\,A| = |A|^{2} = 2^{2} = 4\)
Now take M = adj A, which is also of order 3:
\(|\mathrm{adj}(\mathrm{adj}\,A)| = |\mathrm{adj}\,A|^{2}\)
\(|\mathrm{adj}(\mathrm{adj}\,A)| = 4^{2} = 16\)
In one step: \(|\mathrm{adj}(\mathrm{adj}\,A)| = |A|^{(n - 1)^{2}} = 2^{4} = 16\).
Why the other options are incorrect
Option (1) 4 — Incorrect. This is \(|\mathrm{adj}\,A|\); the rule has to be applied a second time.
Option (2) 8 — Incorrect. This is \(2^{3}\), using the power n = 3 instead of \((n - 1)^{2} = 4\).
Option (3) 64 — Incorrect. This is \(4^{3}\); the power for an order-3 adjoint is 2, not 3.