The number of $3\times 3$  non-singular matrices with four entries as 1 and all other entries as 0 is

The matrix$\left[\begin{array}{ccc}1& a& b\\ c& 1& d\\ e& f& 1\end{array}\right]$  where exactly one of a $a,b,c,d,e,f$  is 1and rest of them are zeros is invertible. There are six such six matrices. Also $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$  is invertible.