Let \( A=\begin{bmatrix}1 & -3\\1 & 2\end{bmatrix} \) \( B=\begin{bmatrix}4 & 6\\-2 & 2\end{bmatrix} \) and \( C \) be a matrix such that \( C=AB \) . Then \( |(BC)^{-1}| \) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0.0005
Since \( |XY|=|X|\,|Y| \) and \( |X^{-1}|=\frac{1}{|X|} \), only the determinants of the factors are needed:
\( |A|=2-(-3)=5 \)
\( |B|=8-(-12)=20 \)
\( |C|=|A|\,|B|=5\times 20=100 \)
\( |BC|=|B|\,|C|=20\times 100=2000 \)
\( \displaystyle |(BC)^{-1}|=\frac{1}{|BC|}=\frac{1}{2000} \)
\( \displaystyle =0.0005 \)