Note Set 05a: A quixotic quest for the inverses of matrices

We explain how and when a square matrix $\mathbf{A}$ can be undone by its inverse $\mathbf{A}^{-1}$, meaning $\mathbf{A}^{-1} \mathbf{A} = \mathbf{I} = \mathbf{A} \mathbf{A}^{-1}$. We give a basic example, talk about identity matrices, and connect to what inverses mean for the number of solutions of $\mathbf{A}\vec{x}=\vec{b}.$ We foreshadow the ethereal realm of the null space of $\mathbf{A}$. Our first method for finding inverses follows in Episode 5b.
Best consumed by 2016/09/12