Nilpotent $4 × 4$ matrix

Question

Find a nilpotent $4 × 4$ matrix with only $-1$ and $3$ as entries.

$\begin{bmatrix} -1 & -1 & 3 & 3 \\ 3 & -1 & 3 & -1 \\ 3 & -1 & 3& -1 \\ -1 & -1 & 3 & 3 \end{bmatrix}$ is the matrix I came up with, but I don't think it works. Anyone have a better one?

Answer 1

$$ \left( \begin{array}{cccc} 3 & 3 & 3 & 3 \\ -1 & -1 & -1 & -1 \\ -1 & -1 & -1 & -1 \\ -1 & -1 & -1 & -1 \\ \end{array} \right) $$

Answer 2

Zero is the only eigenvalue of a nilpotent matrix, so, up to permutation you know that its main diagonal must be $$\begin{pmatrix}3&&&\\&-1&&\\&&-1&\\&&&-1\end{pmatrix}.$$ The matrix must also be singular, which means that it’s going to have linearly dependent columns. Can you fill out the rest of the matrix so that this is true? (There’s more than one way to do that.)