Paragraph: Let A be the set of all 3 3 symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 . Question: The number of matrices A in A for which the system of linear equations A [ array l x y z array ]= [ array l 1 0 0 array ] has a unique solution, is

Step-by-step solution

Given, A [ array l x y z array ]= [ array l 1 0 0 array ] For unique solution, det (A) 0 gathered Case I det (A)= | array lll 1 & a & b a & 1 & c b & c & 1 array | =1-a^2-b^2-c^2+2 a b c 0 gathered Here a, b, c is selected from 1,0,0 . (No case is possible) Case II (i) det (A)= | array lll 1 & a & b a & 0 & c b & c & 0 array |=2 a b c-c^2 0 Here a, b, c are selected from 1,1,0 . (2 cases are possible) (ii) det (A)= | array lll 0 & a & b a & 1 & c b & c & 0 array |=2 a b c-b^2 0 Here a, b, c are selected from 1,1,0