Find the total number of 3×3 non singular matrices with four entries as 1 and all other entries as 0?

Let A be a non-singular matrix with 3 entries as 1 and all other entries as 0.

 

Since A must not have either a zero row or zero column, there are only 2 possibilities of A:

 

 or 

 

To obatin any non-singular matrix with 4 entries as 1 and all other entries as 0, take either matrix and replace any 0 with a 1.

 

Since there are 6 zeros in each matrix, and it is not possible for both matrices to output the same matrix after undergoing replacement, total number of 3×3 non singular matrices with four entries as 1 and all other entries as 0 = 6*2 = 12