The MatrixRank of a Matrix A, rank(A) is the VectorSpaceDimension of the RowSpace and ColumnSpace of A.
If A is any Matrix the RowSpace and ColumnSpace have the same VectorSpaceDimension
If A is n*n then
(MatrixDeterminant det(A) != 0) if and only if (rank(A) = n)
For instance paste
1,0,1,1
3,2,5,1
0,4,4,-4
Into http://wims.unice.fr/wims/wims.cgi?session=3G0DBDBD76.5&+lang=en&+module=tool%2Flinear%2Fmatrix.en click "Show" it will return rank 2 (because the VectorSpaceDimension of the RowSpace is 2).
SingularValueDecomposition can be used to calculate the rank of a matrix.