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

The correct option is C

$A+B$

$AB=B\Rightarrow \left(AB\right)A=BA\Rightarrow A\left(BA\right)=BA\Rightarrow A\left(A\right)=A[\because BA=A]\Rightarrow A^2=AAB=B\Rightarrow B\left(AB\right)=BB\Rightarrow \left(BA\right)B=B^2\Rightarrow AB=B^2\Rightarrow B=B^{2}i.e.B^2=B\therefore A^2+B^2=A+B$