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] 

2  A is rowequivalent to I
[i.e. a finite number of elementary row operations will reduce A to I.] 

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). 


The null space of A is {0}. 


For all nx1 matrices B, AX=B has at least one solution. 


For all nx1 matrices B, AX=B has at most one solution. 


det(A) is nonzero. 


The columns of A span R^{n}. 


The columns of A are linearly independent. 


The rows of A are linearly independent. 


A has rank n. 


A has nullity 0. 


A has no zero eigenvalue. 


... many other equivalent properties ... 
A student view of the Megatheorem.
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