## 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).