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.