Given that the $A_{11}$ and $A_{22} \in \mathbb{R}^{3 x 3}$ are invertible, $A_{21} \in \mathbb{R}^{3 x 3}$, and $b_1, b_2, x_1, x_2 \in \mathbb{R}^{3}$, then solve for $x_1$ and $x_2$ from
$\begin{bmatrix} A_{11} & 0\\ A_{21} & A_{22}\\ \end{bmatrix}$ $\begin{align} \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} \end{align} =$ $\begin{align} \begin{bmatrix} b_{1} \\ b_{2} \\ \end{bmatrix} \end{align}$
What are $x_1$ and $x_2$ in terms of $A_{11}, A_{21}, A_{22}, b_1, b_2$?
Split it up into two equations
$$ \begin{aligned} A_{11} x_1 & = b_1 \\ A_{21} x_1 + A_{22} x_2 & = b_2 \end{aligned}$$
Solve for $x_1$ and in the first equation and use it in the second
$$ \begin{aligned} x_1 & = A_{11}^{-1} b_1 \\ A_{22} x_2 & = b_2 - A_{21} A_{11}^{-1} b_1 \\ \end{aligned} $$
Solve for $x_2$
$$ x_2 = A_{22}^{-1} \left(b_2 - A_{21} A_{11}^{-1} b_1 \right) $$
Re-combine the solution
$$ \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} A_{11}^{-1} & 0 \\ -A_{22}^{-1} A_{21} A_{11}^{-1} & A_{22}^{-1} \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} $$