Problem: I'm analyzing an enterprise for the purpose of investment and I need to find out the average cost of opening a single store. The struggle here is that the company has 3 different brands, so each store has a different cost to be opened.
Available data: all net opening stores for each brand since 2005 (annual data). Investment made to open new stores since 2005.
I thought about a system, Ax = B, A being the net opening of each brand and B being the investment made, but since A is non-squared, this may have infinite or no solution.
Is there any way to get an approximation for these values?
The classic solution is to use the pseudo-inverse of $A$ denoted by $A^+$ to solve the aystem $x = A^+ B$.
If there is a solution, the pseudo-inverse is equal to the inverse. If there is no solution you get the least squares approximation to the solution, so you get the $x$ which is minimizing the residual of $\| A x - B\|^2$.
If there are multiple solutions you still get one solution which is the least squares approximation but also with the least norm of $\|x\|^2$.
You can always compute the pseudo-inverse even if $A$ is singular. For details on how to compute the pseudo-inverse see:
https://en.m.wikipedia.org/wiki/Moore-Penrose_inverse