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.