The number of 3 3 non-singular matrices, with four entries as 1 and all other entries as 0, is (1) 5 (2) 6 (3) at least 7 (4) less than 4

Options

A. at least 7
B. less than 4
C. 5
D. 6

Answer

A

Explanation

Let `A=[(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))]`
`:.` det. `(A)=a_(1)b_(2)c_(3)+a_(2)b_(3)c_(1)+b_(1)c_(2)a_(3)-a_(3)b_(2)c_(1)-a_(2)b_(1)c_(3)-a_(1)b_(3)c_(2)`
If any of the terms is non-zero, then det. (A) will be non-zero and all the elements of that term will be unity.
Now there are 6 elements remaining out of which any one can by unity.
Hence number of non-singular matrices
= (no. of ways of choosing any one triplet)`xx`(no. of ways choosing any one element)
`=^(6)C_(1)xx.^(6)C_(1)`