Page 82 - A Course in Linear Algebra with Applications

66 Chapter Three: Determinants

then

= an a i2 a a —
M 23 l l 3 2 ^12031
«31 ^32
and
I23 = -1) 2 + 3 M 2 3 = ai 2 a 3 i - ana 3 2 .
(

One reason for the introduction of cofactors is that they
provide us with methods of calculating determinants called
row expansion and column expansion. These are a great im-
provement on the defining sum as a means of computing de-
terminants. The next result tells us how they operate.

Theorem 3.1.4
Let A = (dij) be an n x n matrix. Then
a
(i) det(A) = X]fc=i ikMk , (expansion by row i);
a
(ii) det(A) = Efc i kjAkj, (expansion by column j).
=
Thus to expand by row i, we multiply each element in
row i by its cofactor and add up the resulting products.

Proof of Theorem 3.1.4
We shall give the proof of (i); the proof of (ii) is similar. It
is sufficient to show that the coefficient of a^ in the defining
expansion of det(.A) equals A^. Consider first the simplest
case, where i = 1 = k. The terms in the defining expansion of
a
det(.A) that involve n are those that appear in the sum

Y^ si n 1 *2, ••• , in)ana 2i2 •••a nin.
g ( >

Here the sum is taken over all permutations of 1,2,... ,n which
have the form 1, z 2 ,i 3 ,..., z n. This sum is clearly the same as

a
au(%2 sign(i 2, i 3, ... , i n)a2i 2 3i 3