The invertible matrix theorem is a theorem in linear algebra which gives a series of equivalent conditions for an square matrix to have an inverse. In particular, is invertible if and only if any (and hence, all) of the following hold:
1. is row-equivalent to the identity matrix .
2. has pivot positions.
3. The equation has only the trivial solution .
4. The columns of form a linearly independent set.
5. The linear transformation is one-to-one.
6. For each column vector , the equation has a unique solution.
7. The columns of span .
8. The linear transformation is a surjection.
9. There is an matrix such that .
10. There is an matrix such that .
11. The transpose matrix is invertible.
12. The columns of form a basis for .
13. The column space of is equal to .
14. The dimension of the column space of is .
15. The rank of is .
16. The null space of is .
17. The dimension of the null space of is 0.
18. fails to be an eigenvalue of .
19. The determinant of is not zero.
20. The orthogonal complement of the column space of is .
21. The orthogonal complement of the null space of is .
22. The row space of is .
23. The matrix has non-zero singular values.