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