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Elementary Matrices in Linear Algebra: A Comprehensive Guide, Study notes of Elementary Mathematics

There are three types of elementary matrices, each corresponding to one of the types of elementary row operations. Note that elementary matrices are necessarily ...

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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: P1
ij =Pij
Determinant: det(Pij) = 1
1
pf3

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Math 571

Elementary Matrices

  1. Preliminaries

Consider the following situation: A is a matrix, possible augmented, and U is the reduced row echelon form

of A. The U is obtained from A by 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

A by a sequence of matrices Ei called elementary matrices. In other words, U = Ek · · · E 1 A. 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.

  1. Definitions and Notation

We will assume that all of our matrices are of size n × n and 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 × n identity matrix In with row i and row j interchanged

Examples: P 12 =

P 12 A =

d e f

a b c

g h i

AP 12

b a c

e d f

h g i

Multiplication on Left: Interchanges row i and row j of A.

Multiplication on Right: Interchanges column i and column j of A.

Inverse: P

− 1 ij = Pij

Determinant: det(Pij ) = − 1

  1. Diagonal Matrices

Notation: Di(t), t 6 = 0

Definition: Di(t) = the n × n identity matrix In with t in row i, column i

Examples: D 3 (5) =

D 3 (5)A =

a b c

d e f

5 g 5 h 5 i

AD 3 (5)

a b 5 c

d e 5 f

g h 5 i

Multiplication on Left: Multiplies row i of A by t.

Multiplication on Right: Multiplies column i of A by t

Inverse: Di(t)

− 1 = Di(1/t)

Determinant: det(Di(t)) = t

  1. Unipotent Matrices

Notation: Eij (t), i 6 = j

Definition: Eij (t) = the n × n identity matrix In with t in row i, column j

Examples: E 13 (4) =

E 13 (4)A =

a + 4g b + 4h c + 4i

d e f

g h i

AE 13 (4)

a b 4 a + c

d e 4 d + f

g h 4 g + i

Multiplication on Left: Adds t times row j of A to row i of A.

Multiplication on Right: Adds t times column i of A to column j of A.

Inverse: Eij (t) − 1 = Eij (−t)

Determinant: det(Eij (t)) = 1