Can the inverse of a matrix with all entrances different from zero have zero entrances?

Question

Assume X is an invertible $n \times n$ matrix with all entrances different from zero.

I was wondering: can its inverse have a zero in some entrance?

Answer 1

$$\begin{bmatrix}1&1&1\\-1&1&1\\1&-1&1 \end{bmatrix}$$