The Invertible Matrix Theorem

Theorem. Let $A$ be an $n$-by-$n$ 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 $n$-by-$n$ identity matrix.

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

4. The equation $A\mathbf{x}=\mathbf{0}$ has only the trivial solution.

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

6. The linear transformation $\mathbf{x} \mapsto A\mathbf{x}$ is one-to-one.

7. The equation $A\mathbf{x} = \mathbf{b}$ has at least one solution for each $\mathbf{b} \in \mathbb{R}^n$.

8. The columns of $A$ span $\mathbb{R}^n$.

9. The linear transformation $A \mapsto A\mathbf{x}$ is onto.

10. There is an $n$-by-$n$ matrix $C$ such that $CA = I$.

11. There is an $n$-by-$n$ matrix $D$ such that $AD = I$.

12. $A^T$ is an invertible matrix.

13. The columns of $A$ form a basis of $\mathbb{R}^n$.

14. $\mathrm{Col}\, A = \mathbb{R}^n$.

15. $\mathrm{dim}\, \mathrm{Col}\, A = n$.

16. $\mathrm{rank}\, A = n$.

17. $\mathrm{Nul}\, A = \{\mathbf{0}\}$.

18. $\mathrm{dim}\, \mathrm{Nul}\, A = 0$.