5.6. Computation of any Determinant by the Gauss-Jordan Process

The forward elimination phase of the Gauss-Jordan process can be used to transform a general square matrix A to an upper triangular form U.  With reference to the table of §5.5, it is clear that

        detU= ( ) p c 1 c q detA

where p is the number of times rows were interchanged and { c 1 ,, c q }  are the q nonzero scalars used to multiply rows during the elimination process.  Hence,

        detA= ( ) p c 1 c q detU   = ( ) p c 1 c q u 11 u nn detI           (5.8a)

Note that Axiom 3 enters these considerations only to fix the value of det I.  Hence, axioms 1 and 2 alone is enough to give

        detA=c( A )detI                             (5.8)

where c( A )  is a scalar that can be calculated as in (5.8a).