## How to calculate the variance in the solution to a set of linear equations?

A device measures counts on $\displaystyle I$ spectral channels, which are related to
the concentration of $\displaystyle J$ elements in a sample. Elements as in elements on the
pereodic table, sample as in a piece of material.

A measurement on the device results in a vector of spectral counts $\displaystyle c_i$. These
counts are related to the composition of the sample measured $\displaystyle r_j$.
$\displaystyle c_i$ is realated to $\displaystyle r_j$ using a second order tensor:

$\displaystyle A_{ij}c_i = r_j$

In order to find this relationship all $\displaystyle K=I\times J$ elements of
$\displaystyle A_{ij}$ must be found, thus, $\displaystyle K$ equations are needed to solve for all of $\displaystyle A_{ij}$.

Taking measurements on samples with known $\displaystyle r_j$ (and known distribution of
$\displaystyle r_j$) results in $\displaystyle J$ equations per measurement. Using $\displaystyle I$ known samples
a set of linear equations is formed to solve for the coefficients $\displaystyle A_{ij}$.

$\displaystyle C_{kl}A_k = R_k$
$\displaystyle 0 < k \le K \nonumber$
$\displaystyle 0 < l \le K \nonumber$

$\displaystyle C_{kl}$ contains the experimentally measured counts on the known reference
samples, $\displaystyle R_k$ contains the measured compositions (on other equipment) of
the reference samples, and $\displaystyle A_k$ is $\displaystyle A_{ij}$ mapped onto a first order tensor.
By taking repeated measurements on the same physical samples, the variance of each member of $\displaystyle C_{kl}$ and $\displaystyle R_k$
is calculated. $\displaystyle A_k$ is solved for computationally by inverting $\displaystyle C_{kl}$ and multiplying both sides, using Cramer's rule, row reducing or any number of computational methods. For example:

$\displaystyle A_k = C_{kl}^{-1} R_k$

1. Given that the variance of the members of $\displaystyle C_{kl}$ and $\displaystyle R_{k}$ are known, how can
this information be used to compute the variance of the members of $\displaystyle A_k$?
2. When taking a measurement on a sample with unknown composition (once
$\displaystyle A_{ij}$ has been calculated) how can the variance of $\displaystyle r_j$ be calculated?