Solved – Propagation of uncertainty through a linear system of equations

If I have a system of equations, $Ax=B$ where the elements of $B$ have been experimentally determined and as such each element has some uncertainty, how would I propagate this to the elements of $x$?

a_{11} & a_{12}\
a_{21} & a_{22}\
left[begin{matrix} x_{11}\x_{21}end{matrix}right]=
left[begin{matrix} b_{11}pmsigma_{b_{11}}\b_{21}pmsigma_{b_{21}}end{matrix}right]

For instance, in a system like the one above, how do I account for the error in $B$ when solving for $x$? I am trying to find
$sigma_{x_{11}}$ and $sigma_{x_{12}}$.

Let me translate into statistician. So $B$ is a random variable where $B = beta + varepsilon$, with $text{Var}(varepsilon)$ = $Sigma_B$, for $Sigma_B$ known. An observation is taken, and the observed value of $B$ is $b$.

Assuming $A$ is invertible, the solution of $Ax = b$ is $A^{-1}b$. Let $C = A^{-1}$ for the moment.

$text{Var}(Cb) = C,text{Var}(b),C^top = A^{-1} Sigma_B (A^{-1})^top$

If the two components of $b$ are independent, then $Sigma_B$ is diagonal, with diagonal the squares of your $sigma$'s. That variance-covariance matrix of $x$ is in general not diagonal, meaning the values are correlated. The square roots of the diagonal elements of $ A^{-1} Sigma_B (A^{-1})^top$ are the standard deviations of the components of $x$.

This approach applies to more than two dimensions as well.

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