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 $\left|A^{-1}\cdot adj\left(B^{-1}\right)\cdot adj\left(2A^{-1}\right)\right|$ is equal to (where $adj(M)$ represents the adjoint matrix of $M$ )

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

Answer 1

Answer: (D)

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