Simplify all equations and set the variables equal to zero.

You can eliminate the third and fourth row because it's the 'same' as the first and second when simplified.

You get:

y+x=0

z-w=0

=>

Any matrix [w, x; y, z] (where ; is the next row) can commute with matrix A if y+x=0 and z-w=0 or y=-x and z=w.