If A is a square matrix of order 3 such that \(|A| = 3\), then \(|\mathrm{adj}(\mathrm{adj}\, A)|\) is equal to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 81
For a square matrix of order n, the determinant of the adjoint is
\(|\mathrm{adj}\,A| = |A|^{n-1}\)
Here n = 3:
\(|\mathrm{adj}\,A| = 3^{2} = 9\)
\(\mathrm{adj}\,A\) is also of order 3, so the same result applied to it gives
\(|\mathrm{adj}(\mathrm{adj}\,A)| = |\mathrm{adj}\,A|^{2} = 9^{2} = 81\)
Tip: In one step, \(|\mathrm{adj}(\mathrm{adj}\,A)| = |A|^{(n-1)^2}\); here \(3^{4} = 81\). This result is worth remembering.