The invertible matrix theorem - Inverse of Matrices

Notes:

The Invertible Matrix Theorem states the following:
Let $A$ be a square $n \times n$ matrix. Then the following statements are equivalent. That is, for a given $A$, the statements are either all true or all false.

1. $A$ is an invertible matrix.

2. $A$ is row equivalent to the $n \times n$ identity matrix.

3. $A$ has $n$ pivot positions.

4. The equation $Ax=0$ has only the trivial solution.

5. The columns of $A$ form a linearly independent set.

6. The equation $Ax=b$ has at least one solution for each $b$ in $\Bbb{R}^n$.

7. The columns of $A$ span $\Bbb{R}^n$.

8. The linear transformation $x$→$Ax$ maps $\Bbb{R}^n$ onto $\Bbb{R}^n$.

9. There is an $n \times n$ matrix $C$ such that $CA=I$.

10. There is an $n \times n$ matrix $D$ such that $AD=I$.

There are extensions of the invertible matrix theorem, but these are what we need to know for now. Keep in mind that this only works for square matrices.