Question

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$, then $A^2+B^2$ is equal to

• A. $2AB$
• B. $2BA$
• C. $A+B$
• D. $AB$

Answer 1

The correct option is C

$A+B$

Explanation for the correct option:

Given, $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$.

${\mathrm{A}}^2+{\mathrm{B}}^2=\mathrm{AA}+\mathrm{BB}$

$\Rightarrow {\mathrm{A}}^2+{\mathrm{B}}^2=\mathrm{A}\left(\mathrm{BA}\right)+\mathrm{B}\left(\mathrm{AB}\right)$ $\left[\because AB=B,BA=A\right]$

$\Rightarrow A^2+B^2=\left(AB\right)A+\left(BA\right)B$

$\Rightarrow A^2+B^2=BA+AB$ $\left[\because AB=B,BA=A\right]$

$\Rightarrow A^2+B^2=A+B$ $\left[\because AB=B,BA=A\right]$

Hence, option C is correct.