Page 79 - A Course in Linear Algebra with Applications

3.1: The Definition of a Determinant 63

is obtained from I3 by cyclically permuting the columns, C\ —>
C2 —» C3 —> C\. Permutation matrices are easy to recognize
since each row and each column contains a single 1, while all
other entries are zero.
Consider a permutation i\, 22,..., i n of 1,2,..., n, and let
P be the permutation matrix which has (j,ij) entry equal to
1 for j = 1,2,... , n, and all other entries zero. This means
that P is obtained from I n by rearranging the columns in
the manner specified by the permutation i\,. . . ,i n , that is,

Cj Ci,. Then, as matrix multiplication shows,

X
ix
( \ ( \
2 i 2
\nj \i J
n

Thus the effect of a permutation on the order 1,2,... ,n is
reproduced by left multiplication by the corresponding per-
mutation matrix.

Example 3.1.3

The permutation matrix which corresponds to the permuta-
tion 4, 2, 1, 3 is obtained from I4 by the column replacements
Ci —• C4, C2 —> C2, C3 —» Ci, C4 —> C3. It is

/ 0 0 0 1 \
0 1 0 0
P =
1 0 0 0
Vo 0 1 0/