Leading on from a question on the practicality of inverse matrices used in physical engineering systems which I've asked previously, I have come across an issue with my system where I want to take the inverse of a square Matrix (see below) to solve for a set of other variables but I cannot due to its determinant equalling zero.
In my simple system where $W = F^{-1} {\cdot} R$
Below is the matrix $F$ which I'm having trouble getting an inverse for, so I can't seem to solve my system.
From this, I have the following questions:
- Why do determinants equalling zero dictate non-invertible (or "singular"/"degenerate") matrices?
- In the cases where variables I've inputted into my system lead to determinants equalling zero, is there a work around to still somehow get the inverse of this matrix or otherwise solve the system of equations I'm wanting? (i.e. rearranging rows and columns or using some other fancy matrix identities or mathematics?