If \(A=\begin{bmatrix}3 & 0 & -1\\ 2 & 3 & 0\\ 0 & 4 & 1\end{bmatrix}\), then |adj(adj A)| is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1
Expanding |A| along the first row:
\(|A| = 3(3\cdot 1-0\cdot 4)-0+(-1)(2\cdot 4-3\cdot 0)\)
\(= 9-8 = 1\)
For a matrix of order n, \(|\mathrm{adj}\,A| = |A|^{n-1}\). Applying it twice:
\(|\mathrm{adj}(\mathrm{adj}\,A)| = |\mathrm{adj}\,A|^{n-1} = |A|^{(n-1)^2}\)
With n = 3:
\(|\mathrm{adj}(\mathrm{adj}\,A)| = 1^{4} = 1\)