5.21. Existence of the Determinant Function
Apostle treated Theorem 5.20 as an existence theorem for the
determinant function. However, this
existence was already demonstrated by the expansion by row formulae (5.20) and
(5.34). Therefore, it seems more meaningful
to treat Theorem 5.20 as a formula for an expansion by column.
Assume determinants of order n-1 exist. Let f be a function of the n rows of an n ´ n
satisfies all 3 axioms for a determinant function of order n. Therefore, by induction, determinants
of arbitrary order n exists.
By definition, the minor
is a matrix of order n-1.
satisfies all 3 axioms for a determinant
function of order n-1. Now,
so that f
satisfies axiom 1. Next, consider the
matrix B obtained from A by adding row i to row k. Thus,
are of order n-1, they obey axiom 2. Denoting the rows with the 1st
element truncated by a ~, we have
where the factor
comes from moving
from the kth
to the ith row. Hence,
so that f
satisfies axiom 2. Finally, setting
so that axiom 3 is also satisfied. QED.
The foregoing proof is easily adapted to
. Furthermore (5.38) is a column expansion formula.
This is to be contrasted with the row
expansion formula (5.34).
The proof is by induction.
Skipping the trivial case of
we see that the case
is easily verified using eq(5.1). Next, we assume the theorem is true for case n-1 and try to prove case n.
To begin, let
. Expanding detA and detB by the 1st
column and row expansions, respectively, we have
and B1j are order n-1 matrices, we have