In this section we will connect a number of results we learned about matrices and their properties. We know that matrices are useful in several different contexts. Originally we saw how matrices can be used to express and solve systems of linear equations. We also studied the properties of matrices like their row space, column space and null space. In the next chapter, we will also learn about how matrices can be used to represent linear transformations.

In each of these domains, invertible matrices play a particularly important role. The following theorem is a massive collection of facts about invertible matrices.

Invertible matrix theorem: For an $n \times n$ matrix $A$, the following statements are equivalent:

1. $A$ is invertible.
2. The determinant of $A$ is nonzero $\textrm{det}(A) \neq 0$.
3. The equation $A\vec{x} = \vec{b}$ has exactly one solution for each $\vec{b} \in \mathbb{R}^n$.
4. The equation $A\vec{x} = \vec{0}$ has only the trivial solution $\vec{x}=\vec{0}$.
5. The RREF of $A$ is the $n \times n$ identity matrix.
6. The rank of the matrix is $n$.
7. The rows of $A$ are a basis for $\mathbb{R}^n$.
   • The rows of $A$ are linearly independent.
   • The rows of $A$ span $\mathbb{R}^n$. $\mathcal{R}(A)=\mathbb{R}^n$.
8. The columns of $A$ are a basis for $\mathbb{R}^n$.
   • The columns of $A$ are linearly independent.
   • The columns of $A$ span $\mathbb{R}^n$. $\mathcal{C}(A)=\mathbb{R}^n$.
9. The null space of $A$ contains only the zero vector $\mathcal{N}(A)=\{\vec{0}\}$.
10. The transpose $A^T$ is also an invertible matrix.

This theorem states that for a given matrix $A$, the above statements are either all true or all false.

TODO: proof

[ See Section 2.3 of this page for a proof walkthrough ]
http://www.math.nyu.edu/~neylon/linalgfall04/project1/jja/group7.htm