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

Given $AB=B$ and $BA=A$.We need to find $A^2+B^2$.$A^2=A\cdot A$ Since $A=BA$, substitute one $A$:$A^2=A(BA)=(AB)A$ (by associativity)Since $AB=B$:$A^2=BA=A$ Similarly for $B^2$:$B^2=B\cdot B$ Since $B=AB$, substitute one $B$:$B^2=B(AB)=(BA)B$ Since $BA=A$:$B^2=AB=B$ Therefore, $A^2+B^2=A+B$.