Given $BA = A$ and $AB = B$ and $A$ and $B$ are two square matrices, why does $BA +AB = A+B$ ?
Please explain.
ADDED:
The actual problem is it was given that,$BA = A$ and $AB = B$ hence find the value of $A^2 + B^2$.
I simply utilized the fact that if $BA = A$ and $AB = B$ then $A$ and $B$ are idempotent matrices, hence $A^2 + B^2 = A + B$ but the module solution is something more algebraic they break $A^2 + B^2$ and then performs usual substitution after that they showed the result but I don't understand how $BA +AB = A+B$.