An nxn matrix A is nonsingular if and only if... 1 There is a matrix B such that AB=BA=I  [this is the definition of having an inverse] OR 2 A is row-equivalent to I [i.e. a finite number of elementary row operations  will reduce A to I.] OR 3 The only nx1 matrix X such that AX=0 is X=0 (i.e. the linear  system AX=0 has only the trivial solution). OR 4 The null space of A is {0}. OR 5 For all nx1 matrices B, AX=B has at least one solution. OR 6 For all nx1 matrices B, AX=B has at most one solution. OR 7 det(A) is nonzero. OR 8 The columns of A span Rn. OR 9 The columns of A are linearly independent. OR 10 The rows of A are linearly independent. OR 11 A has rank n. OR 12 A has nullity 0. OR 13 A has no zero eigenvalue. OR 14 ... many other equivalent properties ...
As far as I can tell, this name for this theorem was coined by Dane Johnson of the famous Linear Algebra class of 1990. You can see Dane in a picture of the class, in the back row looking down at his book. How widespread has this name become? If you have heard it elsewhere, please e-mail me!