A question about diagonalizable matrices

Question

Let $A$ be a square matrix such that $A \ne0$, but $A^k=0$ for some integer $k \gt1$. show that $A$ is not diagonalizable.

Could somebody give me some hints?Many thanks

Answer 1

Suppose it is diagonlisable, then $A = \Lambda D\Lambda^{-1}$, then $D^k = 0$ (why?)

then, given $D$ is diagonal, what is $D$, so what is $A$?

Answer 2

If a matrix A is diagonalizable, it means there is some other matrix, let's call it S, such that $D=SAS^-1$ is a diagonal matrix (e.g. it has all nonzero values in the diagonal entries and 0 everywhere else).

But $A^k=0$.

But then $D^k=SAS^-1SAS^-1...SAS^-1SAS^-1=SA^kS^-1=0$.

But $D^k$ can't equal 0, because it is a diagonal matrix.