## List of Nonsingular Equivalences

For any nxn square matrix A, these statements are all equivalent to each other; that is, if you know that one of them is true, you know that ALL of them are true; if you know that one of them is false, you know that ALL of them are false.
• A is nonsingular.
• A has an inverse A-1 such that A·A-1 = In.
• x = 0 has only the trivial solution x = 0.
• A is row equivalent to the identity matrix In.
• x = b has a unique solution.
• det(A) ≠ 0.
• The row space and column space of A are n-dimensional.
• The rank of A is n.
• The null space of A is {0}.
• The nullity of A is 0.
• The rows of A are linearly independent.
• The columns of A are linearly independent.
• Zero is not an eigenvalue of A.
(This is of course not a complete list, but it is a good collection of useful ways of saying the same thing. Largely borrowed from Introductory Linear Algebra with Applications by Kolman.)

A useful exercise, if you want some extra practice, is to make up a matrix that makes one of these statements true, and verify that all the rest are true as well--or make up a matrix that makes one of them false and verify that all the rest are false as well. Give it a try!

Back