How do you determine if a matrix is invertible?
The theorem is basically just a long list of statements which are all equivalent to a square matrix being invertible – so any of them can be used as a test of invertibility. Here are a few of the more useful ones:
- The matrix is row equivalent to the identity matrix
- The determinant of the matrix is nonzero
- The matrix has full rank
- The function x↦Ax is bijective
- 0 is not an eigenvalue of the matrix