I computed the determinant to $D= a^{3} - 27a-54$. When I solve $D=0$, I get $a=6$ and $a=-3$ Now when I plug in these values and bring the matrix to rref, I end up with the identity matrix, both for $a=6$ and $a=-3$. That's confusing because that would mean the matrix $A$ has $rank=3$ for all values of $a$, and that can't be. So what is going on here?
EDIT: I have already looked at similar questions so please don't mark this as duplicate. From those questions I have understood how to solve this, so that is not the problem. I took the route involving determinants but it gave me results that weren't expected. I have checked the algebra quite a few times and it always seems to come down to this, so I'm posting in the hope that someone can tell me what's wrong.