If \( A \) and \( B \) are invertible matrices of order 3. If \( |A|=2 \) and \( \left|(AB)^{-1}\right|=\displaystyle\frac{-1}{6} \), then the value of \( |B| \) is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −3
Since \( |XY|=|X|\,|Y| \) and \( |X^{-1}|=\frac{1}{|X|} \),
\( \displaystyle \left|(AB)^{-1}\right|=\frac{1}{|AB|} \)
\( \displaystyle =\frac{1}{|A|\,|B|}=\frac{1}{2|B|} \)
This equals \( -\frac{1}{6} \), so
\( \displaystyle 2|B|=-6 \)
\( \displaystyle |B|=-3 \)
Why the other options are incorrect
Option (1) 3 — Incorrect. With |B| = 3, \( |(AB)^{-1}|=\frac{1}{6} \); the sign is lost.
Option (3) 2 — Incorrect. With |B| = 2, \( |(AB)^{-1}|=\frac{1}{4} \).
Option (4) 1 — Incorrect. With |B| = 1, \( |(AB)^{-1}|=\frac{1}{2} \).