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 $\left|A^{- 1} a d j B^{- 1} a d j \left(3 A^{- 1}\right)\right|$ is equal to

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

Answer 1

Answer: (B)

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