The Invertible Matrix Theorem

1. $A$ is invertible.
2. $A$ is row equivalent to $I_n$.
3. $A$ has $n$ pivotal columns (all columns are pivotal).
4. $A\mathbf{x}$ = $0$ has only the trivial solution.
5. The columns of $A$ are linearly independent.
6. The equation $A\mathbf{x}$ = $\mathbf{b}$ has a solution for all $b\in {\mathbb{R}}^n$
7. The columns of $A$ span ${\mathbb{R}}^n$
8. There is a $n\times n$ matrix $C$ so that $CA=I_n$ ($A$ has a left inverse.)
9. There is a $n\times n$ matrix $D$ so that $AD=I_n$ ($A$ has a right inverse.)
10. $A^T$ is invertible

Example:

Is \left[ \begin{array} \\ 1 & 0 &-2\\3&1&-2 \\0&-1&-1 \end{array} \right] invertible? \text{RREF}\left(\left[ \begin{array} \\ 1 & 0 &-2\\3&1&-2 \\0&-1&-1 \end{array} \right]\right) = $I_3$

Every column is pivotal.

So, \left[ \begin{array} \\ 1 & 0 &-2\\3&1&-2 \\0&-1&-1 \end{array} \right] is invertible!