When are multiplication on matrices commutative?

1. According to me multiplication on matrices are commutative only when
   (i) The given matrices are equal
   (ii) When the matrices are diagonal matrices and of same order.
   (iii) When a suitable identity matrix is being used as prefactor or postfactor

Are there any other possibilities when multiplication of matrices commutative?

2.If there is a matrix A and if assume $A^m$ is equal to B then

$A^n*A=B=A*A^n$
How is this possible?Is there any proof for it? What logic is being used here?