Consider the following matrix:
$\begin{bmatrix} 2x & -2y & 0 \\\ 0 & z & y \\ z & 0 & x \\ y & x & 0 \\ 2x & 2y & 2z\end{bmatrix}$
How can we prove that this matrix has always rank $3$ whenever $(x,y,z)$ satisfy $x^{2}+y^{2}+z^{2}=1$? i.e they lie on the unit sphere. (without using row echelon form as it seems messy)