Let A be the set of all `3 xx 3` symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices in A is

Options

A. 12
B. 6
C. 9
D. 3

Answer

A

Explanation

Let the matrix be
`[(a,b,c),(p,q,r),(x,y,z)]`
We have five entries as 1 and remaining four entries as 0. Since matrix is symmetric, we must have even number of zeros for `inej`. We have two cases.
(i) Two entries in diagonal are zero. We can select two places from three (in diagonal) in `.^(3)C_(2)` ways. Now we have to select elements for upper triangle. For upper triangle, we have three places of which one entry is '0' and are '1'. one place from three can be selected in `.^(3) C_(1)` ways. Hence, the number of matrices is `.^(3)C_(2)xx.^(3)C_(1)=9`.
(ii) If all the entries in the principal diagonal are 1, we have two '0' and one '1' in upper triangle. Hence, the number of matrices is 3. therefore, total number of matrices is 12.