If A is a square matrix of order 3 and |A| = 5, then |adj(adj A)| is:

The correct answer is Option (2) → 625

For a square matrix of order n, \(|\mathrm{adj}\,A| = |A|^{n-1}\). With n = 3:

    \(|\mathrm{adj}\,A| = |A|^2 = 25\)

adj A is also of order 3, so applying the same result to it:

    \(|\mathrm{adj}(\mathrm{adj}\,A)| = |\mathrm{adj}\,A|^2\)
    \(= 25^2 = 625\)

Tip: The result worth remembering is \(|\mathrm{adj}(\mathrm{adj}\,A)| = |A|^{(n-1)^2}\); for order 3 this is \(|A|^4\).