A nilpotent matrix is any matrix where such that . I need to prove that if is nilpotent, then is invertible (where is the identity matrix). I am not allowed to use any theorems regarding determinants, trace, linear (in)dependence, eigenvalues, etc. I am only supposed to use basic matrix operations and properties of invertible matrices.
I wrote this proof, but it doesn't feel right, though I can't place my finger on the error.
Proof: Assume for a contradiction that is not invertible; that is, , .
Thus, . Multiplying through (on the right) by gives us .
However, choosing implies that , giving us a contradiction. Therefore is invertible.
Is that a valid proof, or did I screw up somewhere?