Page 156 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
P. 156
Thus
This is a special case of the following general result.
THEOREM 2.3.1
Let , , and be matrices that differ only in a single row, say the th, and assume that the th row of can be
obtained by adding corresponding entries in the th rows of A and . Then
The same result holds for columns.
EXAMPLE 2 Using Theorem 2.3.1
By evaluating the determinants, the reader can check that
Determinant of a Matrix Product
When one considers the complexity of the definitions of matrix multiplication and determinants, it would seem unlikely that
any simple relationship should exist between them. This is what makes the elegant simplicity of the following result so
surprising: We will show that if A and B are square matrices of the same size, then
(2)
The proof of this theorem is fairly intricate, so we will have to develop some preliminary results first. We begin with the
special case of 2 in which A is an elementary matrix. Because this special case is only a prelude to 2, we call it a lemma.
LEMMA 2.3.2
If B is an matrix and is an elementary matrix, then
