The number of 3 × 3 non-singular matrices with four entries as 1 and all other entries as 0, is

at least 7

Taking each leading diagonal entry as 1 and remaining all entries as zero except one which is equal to 1, we get 6 non-singular matrices as given below:

$\begin{bmatrix}1&1&0\\0&1&0\\0&0&1\end{bmatrix},\begin{bmatrix}1&0&1\\0&1&0\\0&0&1\end{bmatrix},\begin{bmatrix}1&0&1\\0&1&1\\0&0&1\end{bmatrix},\begin{bmatrix}1&0&0\\1&1&0\\0&0&1\end{bmatrix},\begin{bmatrix}1&0&0\\0&1&0\\0&1&1\end{bmatrix},\begin{bmatrix}1&0&0\\0&1&0\\1&0&1\end{bmatrix}$

Similarly, we obtain following non-singular matrices

$\begin{bmatrix}0&0&1\\0&1&1\\1&0&0\end{bmatrix},\begin{bmatrix}0&0&1\\1&0&1\\1&0&0\end{bmatrix}$ etc.