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!

A student view of the Mega-theorem.

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Last Modified May 9, 1998.
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