4.1. ELEMENTARY MATRICES 113
=
a11 a12 · · · a1p...
......
aj1 aj2 · · · ajp...
......
ai1 ai2 · · · aip...
......
an1 an2 · · · anp
This has established the following lemma.
Lemma 4.1.3 Let P ij denote the elementary matrix which involves switching the ith andthe jth rows. Then
P ijA = B
where B is obtained from A by switching the ith and the jth rows.
As a consequence of the above lemma, if you have any permutation (i1, · · · , in), itfollows from Lemma 3.3.2 that the corresponding permutation matrix can be obtained bymultiplying finitely many permutation matrices, each of which switch only two rows. Nowevery such permutation matrix in which only two rows are switched has determinant −1.Therefore, the determinant of the permutation matrix for (i1, · · · , in) equals (−1)
pwhere
the given permutation can be obtained by making p switches. Now p is not unique. Thereare many ways to make switches and end up with a given permutation, but what this showsis that the total number of switches is either always odd or always even. That is, you couldnot obtain a given permutation by making 2m switches and 2k+1 switches. A permutationis said to be even if p is even and odd if p is odd. This is an interesting result in abstractalgebra which is obtained very easily from a consideration of elementary matrices and ofcourse the theory of the determinant. Also, this shows that the composition of permutationscorresponds to the product of the corresponding permutation matrices.
To see permutations considered more directly in the context of group theory, you shouldsee a good abstract algebra book such as [18] or [14].
Next consider the row operation which involves multiplying the ith row by a nonzeroconstant, c. The elementary matrix which results from applying this operation to the ith
row of the identity matrix is of the form
1 0. . .
c. . .
0 1
Now consider what this does to a column vector.
1 0. . .
c. . .
0 1
v1...
vi...
vn
=
v1...
cvi...
vn