Invertible Matrix Theorem

$$\begin{array}{c}AB=I\\ ⟹\\ A=B^{-1}\wedge B=A^{-1}\\ ⟹\\ A,B\text{ are invertible }\end{array}$$

Let $A\in {\mathbb{R}}^{n\times n}$. These statements are all equivalent

• $A$ is Invertible
• $A^T$ is invertible.
• $A$ is Row Equivalent to $I_n$.
• $A$ has $n$ Pivots
• $A\vec{x}=0$ has only the trivial solution, i.e. $\vec{x}=0$
• The columns of A are Linearly Indepedent.
• A is Bijective
• The equation $A\vec{x}=\vec{b}$ has a solution $\forall \vec{b}\in {\mathbb{R}}^n$.
• $\text{span }\text{col}A={\mathbb{R}}^n$
• $A$ has a left inverse
• A has a right inverse

• 0 is not an Eigenvalue of A
• $\det A\ne 0$