If A is a square matrix of order 3 such that \(A^{2} = 3A\), then the possible values of \(|A|\) are
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0 or 27
Take the determinant of both sides and use \(|kA| = k^{n}|A|\) for a matrix of order n (here n = 3).
\(|A^{2}| = |3A|\)
\(|A|^{2} = 3^{3}|A| = 27|A|\)
\(|A|\,(|A| - 27) = 0\)
\(|A| = 0\) or \(|A| = 27\)
Both values occur: A = O gives \(|A| = 0\), and A = 3I gives \(A^{2} = 9I = 3A\) with \(|A| = 27\).
Why the other options are incorrect
Option (1) 0 or 3 — Incorrect. This takes \(|3A| = 3|A|\); for order 3 the factor is \(3^{3} = 27\).
Option (2) 0 or 9 — Incorrect. This uses \(3^{2}\), the factor for order 2, not order 3.
Option (3) 27 only — Incorrect. Cancelling \(|A|\) loses the value 0, which A = O shows is possible.