# An invertible matrix minus the diagonal is nilpotent

Let $$A=(a_{ij})$$ be an $$n\times n$$ invertible matrix over $$\mathbb{C}$$ and $$D=diag(a_{11},a_{22},\dots,a_{nn})$$ be the diagonal matrix whoes diagonal entries are same as $$A$$. Suppose $$A-D$$ is nilpotent. Is it true that $$D$$ is invertible?

For $$n=2$$, since every nilpotent matrix with zero diagonal entries is either upper- or lower-triangular, I already know this is true for $$n=2$$.

Thank you!

Thank you for user1551 for giving a counter example for $$n\geq 3$$.

I have modified the question a bit. I would like to assume the matrix $$A$$ having the property that $$a_{ij}\neq 0\Rightarrow a_{ji}=0.$$ Will it be true that $$D$$ is invertible under this assumption?

No. Here is a counterexample for every $$n\ge3$$: $$A=\pmatrix{0&0&1\\ 0&1&-1\\ 1&1&1\\ &&&I_{n-3}} =\pmatrix{0&0&1\\ 0&0&-1\\ 1&1&0\\ &&&0_{n-3}}+\pmatrix{0\\ &1\\ &&1\\ &&&I_{n-3}}.$$ When $$n=3$$, we have $$(A-D)^2=\pmatrix{0&0&1\\ 0&0&-1\\ 1&1&0}^2=\pmatrix{1&1&0\\ -1&-1&0\\ 0&0&0} \text{ and } (A-D)^3=0.$$