Propagation of uncertainty through a linear system of equations, Ax=b, where A and b are correlated I have a linear system of equations in the form Ax = b. The elements of A and b were experimentally determined and as such have some uncertainty. Within each row, A and b are correlated. Between rows, A and b are uncorrelated. How would I propagate this to the elements of x?
\begin{equation}
\begin{bmatrix}
a_{11} \pm \sigma_{a_{11}} & a_{12} \pm \sigma_{a_{12}} & a_{13}  \pm \sigma_{a_{13}} \\  
a_{21} \pm \sigma_{a_{21}} & a_{22} \pm \sigma_{a_{22}} & a_{23} \pm \sigma_{a_{23}} \\
\end{bmatrix}
\begin{bmatrix}
x_{11} \pm \sigma_{x_{11}} \\  
x_{21} \pm \sigma_{x_{21}} \\ 
x_{31} \pm \sigma_{x_{31}} \\ 
\end{bmatrix}
=\begin{bmatrix}
b_{11} \pm \sigma_{b_{11}} \\  
b_{21} \pm \sigma_{b_{21}}
\end{bmatrix}
\end{equation}
, where $x_{11}+x_{21}+x_{31}=1$, and $x_{11}$, $x_{21}$, $x_{21}$ $>0$. I am trying to find $\sigma_{x_{11}}$, $\sigma_{x_{21}}$, and $\sigma_{x_{31}}$.
I have been looking at the threads on propagation of uncertainty through a linear system of equations and variance of dependent variables but I am unable to combine the two ideas.
There has been a similar unanswered question. Will be hugely grateful for any pointers!
Mark Stone has been extremely helpful with a statistical sampling method. If there is a deterministic way, or some understanding of the challenges of a deterministic way, that would also be of interest, possibly not to just myself.
Update: I tested the solution to this problem but it raised a couple of questions on its own.
 A: Basically, the question is how to calculate uncertainties of the derived coefficients in errors-in-variables models. 
One way is to use Monte Carlo simulations, as showed in Mark's answer. For more analytical solutions, you may need to read
https://en.wikipedia.org/wiki/Errors-in-variables_models
and other materials. I haven't used this before. Hope other people can comment on it. You may refer to this post for the difference between uncertainty from MC simulations and uncertainty from analytical derivations:
Is Monte Carlo uncertainty estimation equivalent to analytical error propagation?
By the way, you said that 'Within each row, A and b are correlated'. Of course, A1 and b1 is are correlated, otherwise we won't derive X, right? By stating 'Between rows, A and b are uncorrelated', do you mean your observations of (A1, b1) and (A2, b2) are uncorrelated?
A: Here is a stochastic simulation approach.  I am not ruling out that someone may come up with an approach not requiring simulation.  Nevertheless, the simulation run time should be negligible, so if the goal is a practical solution, this should qualify.
You have a system of 2 linear equations in 3 unknowns, $A x = b$, augmented by the additional linear equation that the sum of the x's equals 1.  This adds a 3rd row to A consisting of all 1's, and a 3rd element of b equal to 1. There is no uncertainty associated with this 3rd row.  Let $A_{augmented}$ be the 2 rows of A augmented by a (3rd row) row of 1's, and $b_{augmented}$ be the 2 elements of b with a 3rd element equal to 1. So your complete system of linear equations is $A_{augmented} x = b_{augmented}$ 
The first step is to either 
1) fit distributions to the values of A and b, accounting for any correlations among the elements, including across A and b, then for each simulation replication, draw a sample of A and b from the fitted distributions (for instance, Multivariate Normal which can allow correlation between A and b) or
2) use raw measurements of A and b "as is", with each measurement (combination between A and b) being used for a simulation replication.
For each of n simulation replications, solve the equation $A_{augmented} x = b_{augmented}$, which provides the simulated value of x for that simulation replication.  Once you have completed the simulation replications, you can assess the error distribution of the n sample values of x (including correlation across elements).  The accuracy you achieve in assessing the uncertainty of x will ultimately be limited by the number m of A and b measurements (decreasing as its square root), and if fitting distributions to A and b, the validity of the distributions you fit. Nevertheless, if using fitted distributions for A and b, you might as well do a large number of replications n, so that you get maximum mileage out of the data you do have - no need for number of replications n to be limited to number of observations m.
