How do you determine if a matrix is invertible?

matrix-is-invertible

#1

How do you determine if a matrix is invertible?

Answer:

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