Let \(A = [a_{ij}]_{2 \times 3}\) and \(B = [b_{ij}]_{3 \times 2}\), then \(|5AB|\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(5^2|AB|\)
A is of order 2 × 3 and B is of order 3 × 2, so AB is a square matrix of order 2.
For a square matrix M of order n, \(|kM| = k^n|M|\). With M = AB, n = 2 and k = 5:
\(|5AB| = 5^2|AB|\)
Why the other options are incorrect
Option (1) \(5^2.\,|A|.\,|B|\) — Incorrect. A and B are not square, so |A| and |B| are not defined. The rule \(|AB| = |A|\,|B|\) holds only for square matrices of the same order.
Option (2) \(5^3.\,|A|.\,|B|\) — Incorrect. |A| and |B| are not defined, and the power of 5 must be the order of AB, which is 2.
Option (4) \(5^3|AB|\) — Incorrect. The power of 5 is the order of AB, which is 2, not 3.