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?