The invertible matrix theorem  Inverse of Matrices
The invertible matrix theorem
Lessons
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.

Intro Lesson
Characterizations of Invertible Matrices Overview:

3.
Understanding the Theorem
Assume that $A$ is a square $n \times n$ matrix. Determine if the following statements are true or false: