Elementary Matrices

Elementary Matrices
MATH 322, Linear Algebra I
J. Robert Buchanan
Department of Mathematics
Spring 2007
J. Robert Buchanan Elementary Matrices

Outline
Today’s discussion will focus on:
elementary matrices and their properties,
using elementary matrices to find the inverse of a matrix (if
the inverse exists),
properties of invertible matrices.
J. Robert Buchanan Elementary Matrices

Elementary Matrices
Definition
An n × n matrix is an elementary matrix if it can be obtained
from In by a single elementary row operation.
Example
E is a 2 × 2 elementary matrix formed by swapping the two
rows of I2.
0
E =
1
1
0
Note the effect it has upon multiplying an arbitrary matrix.
0
1
1 a11
0
a12 a13
a21
=
a22 a23
a21 a22 a23
J. Robert Buchanan Elementary Matrices
a11 a12 a13

Elementary Matrices
Definition
An n × n matrix is an elementary matrix if it can be obtained
from In by a single elementary row operation.
Example
E is a 2 × 2 elementary matrix formed by swapping the two
rows of I2.
0
E =
1
1
0
Note the effect it has upon multiplying an arbitrary matrix.
0
1
1 a11
0
a12 a13
a21
=
a22 a23
a21 a22 a23
J. Robert Buchanan Elementary Matrices
a11 a12 a13

Left Multiplication by E
Theorem
If E is an elementary matrix obtained from Im by performing a
certain elementary row operation and if A is an m × n matrix
then EA is the matrix that results from performing the same
elementary row operation on A.
Example
Let A =
E2 =
⎡
⎣
⎡
⎣
1 2 3
4 5 6
7 8 9
1 1 0
0 1 0
0 0 1
and calculate E1A, E2A, and E3A.
⎤
⎡
⎦ , E1 = ⎣
⎤
⎡
⎦ , E3 = ⎣
J. Robert Buchanan Elementary Matrices
1 0 0
0 2 0
0 0 1
1 0 0
0 0 1
0 1 0
⎤
⎦ ,
⎤
⎦ ,

Left Multiplication by E
Theorem
If E is an elementary matrix obtained from Im by performing a
certain elementary row operation and if A is an m × n matrix
then EA is the matrix that results from performing the same
elementary row operation on A.
Example
Let A =
E2 =
⎡
⎣
⎡
⎣
1 2 3
4 5 6
7 8 9
1 1 0
0 1 0
0 0 1
and calculate E1A, E2A, and E3A.
⎤
⎡
⎦ , E1 = ⎣
⎤
⎡
⎦ , E3 = ⎣
J. Robert Buchanan Elementary Matrices
1 0 0
0 2 0
0 0 1
1 0 0
0 0 1
0 1 0
⎤
⎦ ,
⎤
⎦ ,

Inverse Operations
Every elementary row operation has an inverse elementary row
operation.
Operation Inverse
Multiply row i by c = 0 Multiply row i by 1/c
Swap rows i and j Swap rows i and j
Add c times row i to row j Add −c times row i to row j
Example
⎡
Let E1 = ⎣
1 0 0
0 2 0
0 0 1
⎤
⎡
⎦ , E2 = ⎣
1 1 0
0 1 0
0 0 1
⎤
⎡
⎦, E3 = ⎣
and find the corresponding inverse operations.
J. Robert Buchanan Elementary Matrices
1 0 0
0 0 1
0 1 0
⎤
⎦ ,

Inverse Operations
Every elementary row operation has an inverse elementary row
operation.
Operation Inverse
Multiply row i by c = 0 Multiply row i by 1/c
Swap rows i and j Swap rows i and j
Add c times row i to row j Add −c times row i to row j
Example
⎡
Let E1 = ⎣
1 0 0
0 2 0
0 0 1
⎤
⎡
⎦ , E2 = ⎣
1 1 0
0 1 0
0 0 1
⎤
⎡
⎦, E3 = ⎣
and find the corresponding inverse operations.
J. Robert Buchanan Elementary Matrices
1 0 0
0 0 1
0 1 0
⎤
⎦ ,

Invertibility
Theorem
Every elementary matrix is invertible and the inverse is also an
elementary matrix.
Proof.
J. Robert Buchanan Elementary Matrices

Invertibility
Theorem
Every elementary matrix is invertible and the inverse is also an
elementary matrix.
Proof.
J. Robert Buchanan Elementary Matrices

Equivalence Result
Theorem
If A is an n × n matrix, then the following statements are
equivalent:
1 A is invertible.
2 Ax = 0 has only the trivial solution.
3 The reduced row echelon form of A is In.
4 A is expressible as the product of elementary matrices.
J. Robert Buchanan Elementary Matrices

Equivalence Result
Theorem
If A is an n × n matrix, then the following statements are
equivalent:
1 A is invertible.
2 Ax = 0 has only the trivial solution.
3 The reduced row echelon form of A is In.
4 A is expressible as the product of elementary matrices.
Proof.
1 =⇒ 2
J. Robert Buchanan Elementary Matrices

Equivalence Result
Theorem
If A is an n × n matrix, then the following statements are
equivalent:
1 A is invertible.
2 Ax = 0 has only the trivial solution.
3 The reduced row echelon form of A is In.
4 A is expressible as the product of elementary matrices.
Proof.
1 =⇒ 2 =⇒ 3
J. Robert Buchanan Elementary Matrices

Equivalence Result
Theorem
If A is an n × n matrix, then the following statements are
equivalent:
1 A is invertible.
2 Ax = 0 has only the trivial solution.
3 The reduced row echelon form of A is In.
4 A is expressible as the product of elementary matrices.
Proof.
1 =⇒ 2 =⇒ 3 =⇒ 4
J. Robert Buchanan Elementary Matrices

Equivalence Result
Theorem
If A is an n × n matrix, then the following statements are
equivalent:
1 A is invertible.
2 Ax = 0 has only the trivial solution.
3 The reduced row echelon form of A is In.
4 A is expressible as the product of elementary matrices.
Proof.
1 =⇒ 2 =⇒ 3 =⇒ 4 =⇒ 1
J. Robert Buchanan Elementary Matrices

Row Equivalence
Definition
If matrix B can be obtained from matrix A by a finite sequence
of elementary row operations then A and B are said to be row
equivalent.
Remark: An n × n matrix A is invertible if and only if A is row
equivalent to In.
J. Robert Buchanan Elementary Matrices

Row Equivalence
Definition
If matrix B can be obtained from matrix A by a finite sequence
of elementary row operations then A and B are said to be row
equivalent.
Remark: An n × n matrix A is invertible if and only if A is row
equivalent to In.
J. Robert Buchanan Elementary Matrices

Matrix Inversion Algorithm
Algorithm: to find the inverse of an invertible matrix A, find the
set of elementary row operations which reduces A to I and then
perform this same sequence of operations on I to produce A −1 .
Example
⎡
Find the inverse of A = ⎣
8 1 5
2 −7 −1
3 4 1
⎤
⎦.
J. Robert Buchanan Elementary Matrices

Matrix Inversion Algorithm
Algorithm: to find the inverse of an invertible matrix A, find the
set of elementary row operations which reduces A to I and then
perform this same sequence of operations on I to produce A −1 .
Example
⎡
Find the inverse of A = ⎣
8 1 5
2 −7 −1
3 4 1
⎤
⎦.
J. Robert Buchanan Elementary Matrices

Example
Example
⎡
Find the inverse of A = ⎣
2 −1 0
4 5 −3
1 −4 3
2
⎤
⎦.
J. Robert Buchanan Elementary Matrices

Homework
Read Section 1.5 and work exercises 1–6, 8, 9, 11, 15.
J. Robert Buchanan Elementary Matrices