I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).

$$\left[\begin{array}{cccccccccc}
 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0\\
 x\_{1}^{2} & Nx\_{1} & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
 0 & c & 0 & 0 & 0 & 0 & 0 & -x^2\_{1} & 0 & 0\\
 x\_{2}^{2} & 0 & Nx\_{2} & 0 & 0 & -1 & 0 & 0 & 0 & 0\\
 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & -x^2\_{2} & 0\\
 0 & 0 & 0 & 0 & z\_1 & 0 & 0 & -1 & 1 & 0\\
 x\_{3}^{2} & 0 & 0 & Nx\_{3} & 0 & 0 & -1 & 0 & 0 & 0\\
 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & -x^2\_{3}\\
 0 & 0 & 0 & 0 & z\_2 & z\_2 & 0 & 0 & -1 & 1
\end{array}\right]$$

where all variables are strictly positive, and $\sum x\_i=1$.

Given that it is sparse, one approach that we considered is to do row-reductions and rearranging to reduce it to a block matrix. This is possible and yields:

$$\left[\begin{array}{ccc}
A & B & 0 \\
0 & C & D \\
E & 0 & F\end{array}\right]=\left[\begin{array}{ccc|ccc|ccc}
 N & 0 & 0 & x\_{1}-1 & x\_{1} & x\_{1} & 0 & 0 & 0\\
 0 & N & 0 & x\_{2} &x\_2-1 & x\_{2} & 0 & 0 & 0\\
 0 & 0 & N & x\_{3} & x\_{3} & x\_3-1 & 0 & 0 & 0\\ \hline
 0 & 0 & 0 & x\_1z\_1 & 0 & 0 & -1 & 1 & 0\\
 0 & 0 & 0 & x\_1z\_2 & x\_2z\_2 & 0 & 0 & -1 & 1\\
 0 & 0 & 0 & x\_1 & x\_2 & x\_3 & 0 & 0 & 0\\ \hline
 c & 0 & 0 & 0 & 0 & 0 & -x\_{1}^2 & 0 & 0\\
 0 & c & 0 & 0 & 0 & 0 & 0 & -x\_{2}^2 & 0\\
 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & -x\_{3}^2
\end{array}\right]$$