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.
- A·x = 0 has only the trivial solution x = 0.
- A is row equivalent to the identity matrix In.
- 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!
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