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

at least 7

The correct answer is option (2) : 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

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

So option (2) is correct.