Can an alternating sign matrix that is not a permutation matrix be non-singular?

Question

An alternating sign matrix is a $n\times n$ matrix with entries in the set $\{-1,0,1\}$ such that for each row and column, the non-zero entries alternate between $1$ and $-1$, starting and ending with a $1$. Permutation matrices are alternating sign matrices with no $-1$ entries. The first non-permutation alternating sign matrix is $$ \left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & -1 & 1\\ 0 & 1 & 0\\ \end{array}\right). $$

Can an alternating sign matrix that is not a permutation matrix be non-singular?

Answer 1

$$\begin{pmatrix}0&1&0&0\\1&-1&1&0\\0&1&-1&1\\0&0&1&0\end{pmatrix}$$