## 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} = I_{n}.
- A·
**x** = **0** has only the trivial solution **x** = **0**.
- A is row equivalent to the identity matrix I
_{n}.
- A·
**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