Consider the linear equation $Ax=b$ with $A \in \mathbb{R}^{(n_1+n_2)\times(n_1+n_2)}$, $b \in \mathbb{R}^{n_1+n_2}$, $x=[x_1,x_2]^T \in \mathbb{R}^{n_1+n_2}$, $x_1 \in \mathbb{R}^{n_1}$ and $x_2 \in \mathbb{R}^{n_2}$. It's well-konwn that this equation has a uniqueness solution if $\rm{rank}(A)=n_1+n_2$.
Now, we are only interested about the convergence of $x_1$, i.e., $x_1 \to x_1^*$. The question is what condition should be imposed on the matrix $A$? Or, any other consideration should be made in addition?