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

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

Grade:12

2 Answers

Clement Yung
19 Points
6 years ago
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:
 
\bigl(\begin{smallmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{smallmatrix}\bigr) or \bigl(\begin{smallmatrix} 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{smallmatrix}\bigr)
 
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
Vikas TU
14149 Points
6 years ago
Try all the possibilities
First row with exactly one zero; total number of cases = 6
          First row 2 zeros we get more cases
        So we get atleast 7 such matrices.

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