If \( A=\begin{bmatrix}x+4 & 3\\2 & 1\end{bmatrix} \), \( B=\begin{bmatrix}5 & 2x-1\\2 & x\end{bmatrix} \) and \( |\mathrm{adj}\,(AB)|=32 \), then the value of \( x \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \pm 6 \)
For a 2 × 2 matrix, \( |\mathrm{adj}\,M|=|M|^{2-1}=|M| \), and \( |AB|=|A|\,|B| \).
\( \displaystyle |A|=(x+4)-6=x-2 \)
\( \displaystyle |B|=5x-2(2x-1)=x+2 \)
\( \displaystyle |\mathrm{adj}(AB)|=|AB|=(x-2)(x+2)=x^2-4 \)
\( \displaystyle x^2-4=32 \)
\( \displaystyle x^2=36\ \Rightarrow\ x=\pm 6 \)
Shortcut: Substitute the options into \( x^2-4 \): \( \pm 3 \) gives 5, \( \pm 4 \) gives 12, \( \pm 6 \) gives 32 and \( \pm 2 \) gives 0.