Is there a way to factor a matrix into two matrices with this special property? (see details)

You have a 2×2 matrix M. Is there a way to factor the matrix into two matrices A and B so that M = AB = BA? All matrices should consist of non-negative integers only.

For example, the matrix
[21, 30]
[60, 51]
can be factored into
[1, 4]
[8, 5]
and
[5, 2]
[4, 7]

Any links to relevant sources will be welcomed.

The reason I am interested with this is that if I could factor a matrix into a “prime factorization” that could not be factored any more, then those “prime” factors could be multiplied in any order and I would end up with the original matrix. I think that any matrix would only have one factorization like this, making the factorization unique. Being able to do this is being able to truly factor matrices.

This is NOT homework.