# The invertible matrix theorem

### The invertible matrix theorem

#### Lessons

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.
• 1.
Characterizations of Invertible Matrices Overview:
a)
The Invertible Matrix Theorem
• only works for $n \times n$ square matrices
• If one is true, then they are all true
• If one is false, then they are all false

b)
How to apply the Invertible Matrix Theorem
• Showing a Matrix is invertible
• Shortcuts to know certain statements

• 2.
Showing a Matrix is invertible or not invertible
Is the following matrix invertible?

• 3.
Is the following matrix invertible? Use as few calculations as possible.

• 4.
Understanding the Theorem
Assume that $A$ is a square $n \times n$ matrix. Determine if the following statements are true or false:
a)
If $A$ is an invertible matrix, then the linear transformation $x$$Ax$ maps $\Bbb{R}^n$ onto $\Bbb{R}^n$.

b)
If there is an $n \times n$ matrix $C$ such that $CA=I$, then there is an $n \times n$ matrix $D$ such that $AD=I$

c)
If the equation $Ax=0$ has only the trivial solution, then $A$ is not invertible.

d)
If the equation $Ax=0$ has a non-trivial solution, then $A$ has less than $n$ pivots.

• 5.
Can a square matrix with two identical rows be invertible? Why or why not?

• 6.
Let $A$ and $B$ be $n \times n$ matrix. Show that if $AB$ is invertible, so is $B$.