Do matrices commutate under self multiplication? Suppose we have any matrix $A$, then $A^n = A.A^{n-1} = A^{n-1}.A$ Will this hold for any matrix $A$ or do we have to have some restrictions on matrix $A$? More generally, is this true- $A^{n+m} = A^n.A^m = A^m.A^n$ for any matrix $A$?

In order to be able to find $A^2$ we need $A$ to be a square matrix that is $n\times n $ matrix.

If the matrix is a square matrix, then the multiplication of powers of $A$ is commutative and $$A^{n+m} = A^n.A^m = A^m.A^n$$