Question

If $A$ and $B$ are square matrices of order $3$ such that $\left|A\right|=3$ and $\left|B\right|=2$ , then the value of $\left|A^{-1}adj{B}^{-1}adj\left(3A^{-1}\right)\right|$ is equal to

• A. $27$
• B. $\frac{27}{4}$
• C. $\frac{1}{108}$
• D. $\frac{1}{4}$

Answer 1

Answer: (B)

$\left|A^{-1}adj{B}^{-1}adj\left(3A^{-1}\right)\right|={\left(\left|A\right|\right)}^{-1}\left|adj{B}^{-1}\right|\left|adj\left(3A^{-1}\right)\right|$
$=$ $\frac{1}{|A|}{\left|B^{-1}\right|}^2\times {\left|3A^{-1}\right|}^2$
$=$ $\frac{1}{|A|}\times \frac{1}{{|B|}^2}\times \frac{3^6}{{|A|}^2}$
$=$ $\frac{3^6}{3^3\times 2^2}=\frac{27}{4}$