If A and B are 2 × 2 matrices such that |AB| = 24, |adj A| = 6, then \( |\mathrm{B}^{-1}| \) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1/4
For a square matrix of order n, \( |\mathrm{adj}\,\mathrm{A}|=|\mathrm{A}|^{n-1} \). Here n = 2, so
\( \displaystyle |\mathrm{A}|=|\mathrm{adj}\,\mathrm{A}|=6 \)
\( \displaystyle |\mathrm{AB}|=|\mathrm{A}|\,|\mathrm{B}|\ \Rightarrow\ 24=6|\mathrm{B}| \)
\( \displaystyle |\mathrm{B}|=4 \)
\( \displaystyle |\mathrm{B}^{-1}|=\frac{1}{|\mathrm{B}|}=\frac{1}{4} \)
Why the other options are incorrect
Option (1) 4 — Incorrect. 4 is |B|; the question asks for \( |\mathrm{B}^{-1}| \).
Option (3) 3/2 — Incorrect. \( \frac{3}{2} \) is \( \frac{|\mathrm{A}|}{|\mathrm{B}|} \), not \( |\mathrm{B}^{-1}| \).
Option (4) 1/6 — Incorrect. \( \frac{1}{6} \) is \( |\mathrm{A}^{-1}| \), not \( |\mathrm{B}^{-1}| \).