The number of ordered pairs ( x , y ) for which A = ( array lll 1 & 2 & 1 2 & 2 & x y & 1 & 2 array ) is a singular and symmetric matrix is

Step-by-step solution

Given A= [ array lll 1 & 2 & 1 2 & 2 & x y & 1 & 2 array ] Since A is singular i.e. |A|=0 aligned & | array lll 1 & 2 & 1 2 & 2 & x y & 1 & 2 array |=0 & 1(4-x)-2(4-x y)+1(2-2 y)=0 & 4-x-8+2 x y+2-2 y=0 aligned -2-x-2 y+2 x y=0 ...(i) -2-x-2 y+2 x y=0 Since A is symmetric aligned & A^T=A & [ array lll 1 & 2 & y 2 & 2 & 1 1 & x & 2 array ]= [ array lll 1 & 2 & 1 2 & 2 & x y & 1 & 2 array ] aligned Comparing above, we get y=1, x=1 ...(i)