Why is $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ not diagonalizable

Assume it is diagonalisable the eigenvalue being $0$ with multiplicity $2$. This means that there exists an invertible matrix $P$ such that

$$\begin{bmatrix}0 & 1\\0 &0\end{bmatrix}=P\cdot\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}\cdot P^{-1}=\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}$$

A contradiction

You can check that $A^2 = 0$. If $A$ were diagonalizable, what would then be the options for values on the diagonal?