Question

Let $A$ and $B$ be two non singular matrices of same order such that $\left(A B\right)^{k}=A^{k}B^{k}$ for consecutive positive integral values of $k$ , then $AB^{2}A^{- 1}$ is equal to

• A. $A^{2}$
• B. $B$
• C. $A$
• D. $B^{2}$

Answer 1

Answer: (D)

$\left|A\right|\neq 0,\left|B\right|\neq 0$ (given)
$\left(A B\right)^{3}=A^{3}B^{3}\ldots \left(i\right)$
Also, $\left(A B\right)^{3}=\left(A B\right)^{2}\left(A B\right)=A^{2}B^{2}AB\ldots \left(i i\right)$
From $\left(i\right)$ and $\left(i i\right)$
$AB^{2}=B^{2}A$
$AB^{2}A^{- 1}=B^{2}AA^{- 1}=B^{2}$