The Invertible Matrix Theorem states the following:
be a square
matrix. Then the following statements are equivalent. That is, for a given
, the statements are either all true or all false.
is an invertible matrix.
is row equivalent to the
4. The equation
has only the trivial solution.
5. The columns of
form a linearly independent set.
6. The equation
has at least one solution for each
7. The columns of
8. The linear transformation
9. There is an
10. There is an
There are extensions of the invertible matrix theorem, but these are what we need to know for now. Keep in mind that this only works for square