If a is a matrix of 0, then |B| certainly cannot be equal to 0, because the result of multiplying a matrix of 0 by any matrix is still a matrix of 0.

If a is not a matrix of 0, then |B| must be equal to 0.

By reducing to absurdity, let |B|≠0, then b must be a full rank matrix, that is, the inverse matrix.

Let c be the inverse matrix of b.

Then A=ABC=0*C=0.

This contradicts that the matrix A is not 0'

So if a is not a matrix of 0, then |B|=0.