Question:
If $A$ and $B$ are two matrices such that $A B=B$ and $B A=A, A^{2}+B^{2}$ is equal to
(a) $2 A B$
(b) $2 B A$
(c) $A+B$
(d) $A B$
Solution:
(c) $A+B$
Given : $A B=B$ and $B A=A$
$A^{2}+B^{2}=A A+B B$
$\Rightarrow A^{2}+B^{2}=B A B A+A B A B \quad[\because A B=B$ and $B A=A]$
$\Rightarrow A^{2}+B^{2}=B B A+A A B \quad[\because A B=B$ and $B A=A]$
$\Rightarrow A^{2}+B^{2}=B A+A B \quad[\because A B=B$ and $B A=A]$
$\Rightarrow A^{2}+B^{2}=A+B \quad[\because A B=B$ and $B A=A]$