7)$a11, a12 ,a13, a21,a22,a23, a31,a32,a33$
if all diagonal elements are 1,1,1 then no of remaining elements for the matrix (a12,a13,a23)=(a21,a31,a32)= (0,0,1),(0,1,0),(1,0,0)
so 3 symmetric matrices.
if all diagonal elements are 1,0,0 then no of remaining elements for the matrix (a12,a13,a23)=(a21,a31,a32)=(0,1,1),(1,1,0),(1,0,1)
so 3 symmetric matrices.
if all diagonal elements are 0,0,1 then no of remaining elements for the matrix (a12,a13,a23)=(a21,a31,a32)=(0,1,1),(1,1,0),(1,0,1)
so 3 symmetric matrices.
if all diagonal elements are 0,1,0 then no of remaining elements for the matrix (a12,a13,a23)=(a21,a31,a32)=(0,1,1),(1,1,0),(1,0,1)
so 3 symmetric matrices.
so total 12 symmetric matrices.
8)for diagonal elements (1,1,1) only system is having infinite no of solutions.
i.e) x=1,y+z=0--------if z is any real number then y(=-z) also any real number.so infinite solutions.
so atleast 4 and less than 7
9)all 12 matrices are consistent(9 having unique solutions and 3 having infinite no of solutions)