Question

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

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

Answer 1

Answer: (C)

$A+B$

It is given that $AB=B$ and $BA=A$

To find $A^2+B^2$

Consider $A^2+B^2=AA+BB$

$=BABA+ABAB$

$=BBA+AAB$ (as $AB=B$ and $BA=A$)

$=BA+AB$ (as $BA=A$ and $AB=B$)

$=A+B$ (as $BA=A$ and $AB=B$)

Therefore, $A^2+B^2=A+B$

Hence option $C$ is correct.