Symmetric Matrices & Quadratic Forms · Hard problem

Lagrange's method reduces $q(x,y,z) = x^2-4xy+3y^2+2yz-z^2$ to $(x-2y)^2 - (y-z)^2$ (a sum of only TWO independent squares, even though $q$ has three variables). This signals that $q$'s matrix is singular. Find the RANK of $q$'s symmetric matrix.