is a square matrix such that =0. Prove that for any real number the matrix is invertible

Attempt: if =0 then is a singular matrix. A singular matrix times is still a singular matrix. so the idea is to prove that is an invertible matrix when A is singular.

I'm not sure where to go from here but this is something interesting I found:

C = D = C and D are both singular matrices.

1) C + = = invertible matrix

however,

2) D + = = singluar matrix

So I believe the idea is to figure out why D is not part of the matrix we are looking for, even though it is singular, and it probably has to do with the fact that

Thanks!