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

A device measures counts on $I$ spectral channels, which are related to
the concentration of $J$ 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 $c_i$. These
counts are related to the composition of the sample measured $r_j$.
$c_i$ is realated to $r_j$ using a second order tensor:

$
A_{ij}c_i = r_j
$

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

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

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

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

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

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