If \(A = \begin{bmatrix}1&2\\0&3\end{bmatrix}\), then \(|A\, \mathrm{adj}\,A|\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 9
For a square matrix of order n, \(A(\mathrm{adj}\,A) = |A|\,I_n\). Taking determinants, with \(|kI_n| = k^n\):
\(|A\,\mathrm{adj}\,A| = |A|^n\)
Here \(n = 2\) and
\(|A| = 1 \times 3 - 2 \times 0 = 3\)
so
\(|A\,\mathrm{adj}\,A| = 3^2 = 9\)
Tip: \(|A\,\mathrm{adj}\,A| = |A|^n\), while \(|\mathrm{adj}\,A| = |A|^{n-1}\); for this 2 × 2 matrix they are 9 and 3, and 3 is also an option.