Math 571
Elementary Matrices
1. Preliminaries
Consider the following situation: Ais a matrix, possible augmented, and Uis the reduced row echelon form
of A. The Uis obtained from Aby a series of elementary row operations. However, these operations are, in
some sense, external to the matrix A. It turns out that we can accomplish this row reduction by multiplying
Aby a sequence of matrices Eicalled elementary matrices. In other words, U=Ek· · · E1A. There are three
types of elementary matrices, each corresponding to one of the types of elementary row operations. Note that
elementary matrices are necessarily n×n.
2. Definitions and Notation
We will assume that all of our matrices are of size n×nand our notation will not refer to their size. Also all
specific examples will be 3 ×3 matrices. For convenience let
A=
a b c
d e f
g h i
1. Permutation Matrices
Notation: Pij, i 6=j
Definition: Pij = the n×nidentity matrix Inwith row iand row jinterchanged
Examples: P12 =
010
100
001
P12A=
d e f
a b c
g h i
AP12
b a c
e d f
h g i
Multiplication on Left: Interchanges row iand row jof A.
Multiplication on Right: Interchanges column iand column jof A.
Inverse: P−1
ij =Pij
Determinant: det(Pij) = −1
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