The simplest proofs use just those notions needed to formulate the theorem ( matrices , polynomials with numeric entries , determinants ) .
The simplest proofs involve technical computations that render somewhat mysterious the fact that they lead precisely to the correct conclusion .
The simplest proofs use just those notions needed to formulate the theorem ( matrices , polynomials with numeric entries , determinants ) , but involve technical computations that render somewhat mysterious the fact that they lead precisely to the correct conclusion .