If A is a square matrix of order \(3 \times 3\) and \(|A|= 4\). The value of \(|(\mathrm{adj}\,A).\,A|\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 64
For any square matrix, \((\mathrm{adj}\,A)A = |A|\,I\). Here
\((\mathrm{adj}\,A)A = 4I\)
For a matrix of order n, \(|kI| = k^n\). With n = 3:
\(|(\mathrm{adj}\,A)A| = |4I| = 4^3 = 64\)
Equivalently, \(|\mathrm{adj}\,A| = |A|^{n-1} = 4^2 = 16\), and \(|(\mathrm{adj}\,A)A| = |\mathrm{adj}\,A|\,|A| = 16 \times 4 = 64\).