Theorem

Let A be an n x n - matrix. Then the following statements are equivalent:

1. A is an invertible matrix.
2. A is row equivalent to I_n.
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 linear transformation T:Rⁿ→Rⁿ; T(x)=Ax is one-to-one.
7. The equation Ax = b has at least one solution for each b in Rⁿ.
8. The columns of A span Rⁿ.
9. The linear transformation T:Rⁿ→Rⁿ; T(x)=Ax is onto.
10. There is an n x n - matrix C such that CA=I_n.
11. There is an n x n - matrix D such that AD=I_n.
12. A^T is an invertible matrix.