Let $N$ be a nilpotent matrix. Prove that $Id - N$ is an invertible matrix.

Question

what I think is we may somehow start with let $N$ be the matrix associated with the linear mappin $F : K^n \to K^n$... But I couldn't proceed

Answer 1

We can also check the singularity of the matrix by checking its zero eigenvalues.

The matrix $dI-N$ is singular if and only if it has zero eigenvalues. Assume $dI-N$ has a zero eigenvalue and $x$ is the associated eigenvector such that $$(dI-N)x=0\Leftrightarrow Nx=dx\Rightarrow N^2x=dNx=d^2x\Rightarrow \cdots\Rightarrow N^kx=d^kx$$ Since $N^k=0$, we have $d^kx=0$ and hence $x=0$, which conflicts with the assumption that $x$ is nonzero.