[Math] Is a square matrix whose diagonal and antidiagonal elements are all zero always singular

Question

Consider an $n\times n$ matrix whose primary and secondary diagonal elements are all zero. Does it necessarily follow that the determinant vanishes for these matrices?

When $n=1,2,3,4$, the matrix is necessarily singular.

Answer 1

For the $4\times 4$ case, you can simplify the determinant a bit:

$$\begin{vmatrix}&a&b&\\c&&&f\\d&&&g\\&h&k&\end{vmatrix}=\begin{vmatrix}a&b\\h&k\end{vmatrix}\begin{vmatrix}c&f\\d&g\end{vmatrix}$$

and one can certainly set things up such that both $2\times 2$ determinants do not evaluate to $0$.