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 $\left|\left(2 A\right)^{2} \left(3 B\right) \left(5 C\right)^{- 1}\right|=\frac{1728}{125}$ , then the value of $n$ is equal to

• A. $2$
• B. $3$
• C. $4$
• D. $6$

Answer 1

Answer: (A)

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