If A is square matrix of order 3 × 3 and \(|\mathrm{adj}\,A| = 64\), then the value of \(|5A|\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\pm 1000\)
For a square matrix of order n, \(|\mathrm{adj}\,A| = |A|^{n-1}\). Here n = 3:
\(|A|^2 = 64\)
\(|A| = \pm 8\)
Also \(|kA| = k^n|A|\), since each of the n rows is multiplied by k:
\(|5A| = 5^3|A| = 125 \times (\pm 8) = \pm 1000\)
Tip: When \(n - 1\) is even, \(|\mathrm{adj}\,A|\) fixes \(|A|\) only up to sign; that is why the answer here carries \(\pm\).