Page 468 - Matrix Analysis & Applied Linear Algebra

464 Chapter 6 Determinants

Proof of (6.1.6). If B agrees with A except that B i∗ = αA i∗ , then for each
permutation p =(p 1 ,p 2 ,...,p n ),
),
b 1p 1 ··· b ip i ··· b np n = a 1p 1 ··· αa ip i ··· a np n = α(a 1p 1 ··· a ip i ··· a np n
and therefore the expansion (6.1.1) yields det (B)= α det (A).
Proof of (6.1.7). If B agrees with A except that B j∗ = A j∗ + αA i∗ , then
for each permutation p =(p 1 ,p 2 ,...,p n ),

b 1p 1 ··· b ip i ··· b jp j ··· b np n = a 1p 1 ··· a ip i ··· (a jp j + αa ip j ) ··· a np n
),
= a 1p 1 ··· a ip i ··· a jp j ··· a np n + α(a 1p 1 ··· a ip i ··· a ip j ··· a np n
so that

det (B)= σ(p)a 1p 1 ··· a ip i ··· a jp j ··· a np n
p
(6.1.8)

+α σ(p)a 1p 1 ··· a ip i ··· a ip j ··· a np n .
p
The first sum on the right-hand side of (6.1.8) is det (A), while the second sum is
˜
the expansion of the determinant of a matrix A in which the i th and j th rows
˜
are identical. For such a matrix, det(A) = 0 because (6.1.5) says that the sign
of the determinant is reversed whenever the i th and j th rows are interchanged,
˜
˜
so det(A)= −det(A). Consequently, the second sum on the right-hand side of
(6.1.8) is zero, and thus det (B) = det (A).
It is now possible to evaluate the determinant of an elementary matrix as-
sociated with any of the three types of elementary operations. Let E, F, and
G be elementary matrices of Types I, II, and III, respectively, and recall from
the discussion in §3.9 that each of these elementary matrices can be obtained by
performing the associated row (or column) operation to an identity matrix of ap-
propriate size. The result concerning triangular determinants (6.1.3) guarantees
that det (I) = 1 regardless of the size of I, so if E is obtained by interchanging
any two rows (or columns) in I, then (6.1.5) insures that

det (E)= −det (I)= −1. (6.1.9)

Similarly, if F is obtained by multiplying any row (or column) in I by α =0,
then (6.1.6) implies that

det (F)= α det (I)= α, (6.1.10)

and if G is the result of adding a multiple of one row (or column) in I to
another row (or column) in I, then (6.1.7) guarantees that

det (G) = det (I)=1. (6.1.11)