Question

Let $A$ and $B$ be two square matrices of order $3$ such that $\left|A\right|=3$ and $\left|B\right|=2$ , then the value of $\left|A^{- 1} \cdot a d j \left(B^{- 1}\right) \cdot a d j \left(2 A^{- 1}\right)\right|$ is equal to (where $adj\left(\right.M\left.\right)$ represents the adjoint matrix of $M$ )

• A. 72
• B. $\frac{64}{27}$
• C. $\frac{8}{9}$
• D. $\frac{16}{27}$

Answer 1

Answer: (D)

$\left|A^{- 1} \cdot a d j \left(B^{- 1}\right) \cdot a d j \left(2 A^{- 1}\right)\right|=\frac{1}{\left|A\right|}\cdot \left(\left|B^{- 1}\right|\right)^{2}\left(\left|2 A^{- 1}\right|\right)^{2}$
$=\frac{1}{\left|A\right|}\times \frac{1}{\left|B\right|^{2}}\cdot \frac{2^{6}}{\left|A\right|^{2}}=\frac{2^{6}}{3^{3} \cdot 2^{2}}=\frac{16}{27}$