Inverse

#Math

$\text{Rules}$

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

1. $\exists A^{-1}$
2. $\exists!\vec x\in\mathbb{R}^n:A\vec x=\vec b\space\forall\space\vec b\in\mathbb{R}^n$
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 $A$