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?