Invertible Matrix Theorem
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.
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