Question

If two matrices $A$ and $B$ are such that, $AB=A$ and $BA=B.$ Then $A^3+B^3$ is equal to

• A. $AB$
• B. $A^3$
• C. $B^3$
• D. $A+B$

Answer 1

The correct option is D $A+B$

Given : $AB=A$
$\Rightarrow A\left(BA\right)=A^2\Rightarrow AB=A^2(\because BA=B)\Rightarrow A\left(BA\right)=A^3\Rightarrow AB=A^3=A$
Similarly, $BA=B$
$\Rightarrow B\left(AB\right)=B^2\Rightarrow BA=B^2(\because AB=A)\Rightarrow B\left(AB\right)=B^3$
$\Rightarrow BA=B^3=B$
So, $A^3+B^3=A+B$