Question

Let $A,B$ and $C$ are $n\times n$ matrices such that $\left|A\right|=2,\left|B\right|=3$ and $\left|C\right|=5.$ If $\left|{\left(2A\right)}^2\left(3B\right){\left(5C\right)}^{-1}\right|=\frac{1728}{125}$ , then the value of $n$ is equal to

• A. $2$
• B. $3$
• C. $4$
• D. $6$

Answer 1

Answer: (A)

$\left|{\left(2A\right)}^2\left(3B\right){\left(5C\right)}^{-1}\right|=\frac{{\left(|2A|\right)}^2|3B|}{|5C|}=\frac{2^{2n}{3}^n}{5^n}\frac{{\left(|A|\right)}^2|B|}{|C|}$
$\Rightarrow {\left(\frac{12}{5}\right)}^{n+1}={\left(\frac{12}{5}\right)}^3\Rightarrow n=2$