# Given that the A_11 and A_22 in RR^(3 xx 3) are invertible, A_21 in RR^(3 xx 3), and b_1,b_2,x_1,x_2 in RR^3, then solve for x_1 and x_2 from [[A_11,0],[A_21,A_22]] [[x_1],[x_2]]=[[b_1],[b_2]] What are x_1 and x_2 in terms of A_11,A_21,A_22,b_1,b_2?

Given that the ${A}_{11}$ and ${A}_{22}\in {\mathbb{R}}^{3x3}$ are invertible, ${A}_{21}\in {\mathbb{R}}^{3x3}$, and ${b}_{1},{b}_{2},{x}_{1},{x}_{2}\in {\mathbb{R}}^{3}$, then solve for ${x}_{1}$ and${x}_{2}$ from
$\left[\begin{array}{cc}{A}_{11}& 0\\ {A}_{21}& {A}_{22}\end{array}\right]$ $\begin{array}{r}\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]\end{array}=$ $\begin{array}{r}\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\end{array}\right]\end{array}$
What are ${x}_{1}$ and ${x}_{2}$ in terms of ${A}_{11},{A}_{21},{A}_{22},{b}_{1},{b}_{2}$?
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Split it up into two equations
$\begin{array}{rl}{A}_{11}{x}_{1}& ={b}_{1}\\ {A}_{21}{x}_{1}+{A}_{22}{x}_{2}& ={b}_{2}\end{array}$
Solve for ${x}_{1}$ and in the first equation and use it in the second
$\begin{array}{rl}{x}_{1}& ={A}_{11}^{-1}{b}_{1}\\ {A}_{22}{x}_{2}& ={b}_{2}-{A}_{21}{A}_{11}^{-1}{b}_{1}\end{array}$
Solve for ${x}_{2}$
${x}_{2}={A}_{22}^{-1}\left({b}_{2}-{A}_{21}{A}_{11}^{-1}{b}_{1}\right)$
Re-combine the solution
$\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{cc}{A}_{11}^{-1}& 0\\ -{A}_{22}^{-1}{A}_{21}{A}_{11}^{-1}& {A}_{22}^{-1}\end{array}\right]\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\end{array}\right]$

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