If A and B are square matrices of order 3 such that \(|A| = -4\) and \(AB = 4I\), then \(|B|\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-16\)
Take determinants of both sides of AB = 4I, using the product rule \(|AB| = |A|\,|B|\) and \(|kI| = k^{3}\) for order 3.
\(|A|\,|B| = |4I| = 4^{3} = 64\)
\(-4\,|B| = 64\)
\(|B| = -16\)
Why the other options are incorrect
Option (2) 16 — Incorrect. Since \(|A|\) is negative and \(|A|\,|B| = 64\) is positive, \(|B|\) must be negative.
Option (3) \(-1\) — Incorrect. This takes \(|4I| = 4\); for order 3 it is \(4^{3} = 64\).
Option (4) \(-64\) — Incorrect. 64 is \(|AB|\); it still has to be divided by \(|A| = -4\).