Let matrix $A\in {\mathbb{R}}^{n\times n}$ and vector $\overrightarrow{b}\in {\mathbb{R}}^n$ be "given." Suppose we want to find the 12.8 Solution Set to Square Linear-Systems Problem $A\overrightarrow{x}=\overrightarrow{b}$. Then, there are only three possible scenarios:

1. Empty solution set: no exact solutions exist
2. One element solution set: a unique solution exists
3. Infinite solution set: non-unique solutions exist

If $A$ is nonsingular, the square linear-systems problem will ALWAYS have a unique solution.