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.
