Page 88 - A Course in Linear Algebra with Applications

72 Chapter Three: Determinants

since a jifc = a,ki k and a^ = a^. This means that the term
under consideration occurs a second time in the denning sum
for det(.A), but with the opposite sign. Therefore all terms in
the sum cancel and det(^4) equals zero.
Notice that we do not need to prove the statement for
columns because of the remark following 3.2.1.

The next three results describe the effect on a determi-
nant of applying a row or column operation to the associated
matrix.

Theorem 3.2.3
(i) If a single row (or column) of a matrix A is multiplied
by a scalar c, the resulting matrix has determinant equal
to c(det(A)).
(ii) If two rows (or columns) of a matrix A are
interchanged, the effect is to change the sign of the
determinant.
(iii) The determinant of a matrix A is not changed if a
multiple of one row (or column) is added to another row
(or column).

Proof
(i) The effect of the operation is to multiply every term in
the sum defining det(A) by c. Therefore the determinant is
multiplied by c.
(ii) Here the effect of the operation is to switch two entries
in each permutation of 1, ,..., n; we have already seen that
2
this changes the sign of a permutation, so it multiplies the
determinant by —1.
(iii) Suppose that we add c times row j to row k of the matrix:
here we shall assume that j < k. If C is the resulting matrix,
then det(C) equals

a
)
^2 sign(ii,..., i n )aiii • • • %•*,- • • • ( ki k + ca jik • • • a nin,