How to find the rank of this matrix?

Question

Find the rank of

$$\begin{pmatrix} 4y+2z & 4x+4 & 2x-1 \\ y+2 & x & -1 \end{pmatrix}$$

where $(x,y,z)$ satisfy $4xy + 2xz + 4y - z = xy + 2x - z = 0$.

Answer 1

if you solve the system $4xy+2xz+4y-z=xy+2x-z=0$ for $x$ you obtain :

$$y=\frac{2x(1-2x)}{2x^2+3x+4},z=\frac{8x(x+1)}{2x^2+3x+4}$$

So we can rewrite the matrix $M$ as :

$$\begin{pmatrix} \frac{24x}{2x^2+3x+4} & 4x+4 & 2x-1 \\ \frac{8(x+1)}{2x^2+3x+4}& x & -1 \end{pmatrix} $$

Via gaussian elimination (and a good amount of algebra) we get the reduced matrix :

$$\begin{pmatrix} 1& 0 & \frac{-4x^4-12x^3-25x^2-24x-16}{8(x^2+8x+4)} \\ 0& 1 & \frac{2x^2+4x-1}{x^2+8x+4} \end{pmatrix} $$

So $Rank(M)=2$.