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).
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SingularValueDecomposition can be used to calculate the rank of a matrix.
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CategoryMath