The invertible matrix theorem states that for any matrix the following statements are always either all true or all false:

• is invertible.
• has pivots
• The columns of are linearly independent
• The columns of span
• The matrix is invertible
• has a unique solution for each in
• The rank of is
• The determinant of is non-zero
• The null space of is 0
• 0 is not an eigenvalue of

Additionally the linear map formed from also have the same truth value

• is invertible
• is bijective.