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 ...

A student view of the Mega-theorem.