Inverse

$\text{Rules}$

For $A=n\times n$, either all hold true or none do:

1. $\exists A^{-1}$
2. $\exists !\vec{x}\in {\mathbb{R}}^n:A\vec{x}=\vec{b}\forall \vec{b}\in {\mathbb{R}}^n$
3. $\text{rref}(A)=I_n$
4. $\text{rank}(A)=n$
5. $\text{im}(A)={\mathbb{R}}^n$
6. $\text{ker}(A)=\{\vec{0}\}$
7. $\text{Column vectors of }A\text{ form a basis of }{\mathbb{R}}^n$
8. $\text{Column vectors of }A\text{ span }{\mathbb{R}}^n$
9. $\text{Column vectors of }A\text{ are linearly independent}$
10. $\det A\ne 0$
11. $0$ not an eigenvalue of $A$