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Elementary Matrices
3
...
1 and 3
...
In Section 3
...
Since matrix multiplication is a
direct extension of matrix-vector multiplication, it should not be surprising that
there is a connection between matrix multiplication, systems of linear equations, and
linear mappings
...
Definition
A matrix that can be obtained from the identity matrix by a single elementary row
Elementary Matrix
operation is called an elementary matrix
...
EXAMPLE 1
E1=
[� �]
is the elementary matrix obtained from I2 by adding the product oft times
the second row to the first-a single elementary row operation
...
£2 =
[� �]
[� �]
is the elementary matrix obtained from I2 by multiplying row 2 by the
non-zero scalar t
...
£3 =
is the elementary matrix obtained from I2 by swapping row 1 and row 2,
and it is the matrix of a reflection over x2
=
x1 in the plane
...
The following theorem tells us that ele
mentary matrices also represent elementary row operations
...
Theorem 1
If A is an n x n matrix and Eis the elementary matrix obtained from In by a certain
elementary row operation, then the product EA is the matrix obtained from A by
performing the same elementary row operation
...
Instead, we illustrate
[
why this works by verifying the conclusion for some simple cases for 3 x 3 matrices
...
Consider the elementary row operation of adding k times row 3 to row 1
...
Then,
0
0
1
[
a11
a12
a1 3
a21
a22
a23
a31
a32
a33
while
EA=
[�
0
k
1
0
0
1
l[
l
R, + kR,
�
[
l
a11 + ka31
a12 + ka32
a13 + ka33
a21
a22
a23
a31
a32
a33
a11
a12
a13
a21
a22
a31
a32
a23 =
a33
[""
a12 + ka32
a1 3 + ka33
a21
a22
a23
a31
a32
a33
+ ka, ,
l
Case 2
...
EXERCISE 1
Verify that Theorem 1 also holds for the elementary row operation of multiplying the
second row by a non-zero constant for 3 x 3 matrices
...
•
•
•
, Ek
...
Call the elementary
matrix corresponding to the first operation E1, the elementary matrix corresponding
to the second operation E2 , and so on, until the final elementary row operation cor
responds to Ek
...
EXAMPLE2
Let A =
[� � �]
...
, Ek such that
Ek··· E1A is the reduced row echelon form of A
...
[� � ]
The first elementary row operation is Rz - 2R1, so E1=
The second elementary row operation is
1
1
_
1 2
...
[� -�][� 1�2] [ _� �][� � !] [� � �l
The third elementary row operation is R1 - R2, so E3
Thus,E3E2E1A
=
=
=
Remark
We know that the elementary matrices in Example
2
must be 2 x
2
for two reasons
...
Second, for the matrix
multiplication E1A to be defined, we know that the number of columns in E1 must be
equal to the number of rows in A
...
EXERCISE2
Let A
=
[� H
Find a sequence of elementacy matrices £1,
...
2,
In the special case where A is an invertible square matrix, the reduced row eche
lon form of A is I
...
Thus, the matrix B
=
=
Ek··· E1 satisfies BA
=
I,
so B is the inverse of A
...
5
...
Second, it shows us that solving a system Ax
by row reducing or by computing x
=
=
b
A-' b yields the same result
...
Moreover, since the
reverse operation of an elementary row operation is another elementary row operation,
the inverse of an elementary matrix is another elementary matrix
...
Theorem 3
If an n x n matrix A has rank n, then it may be represented as a product of elementary
matrices
...
Since E k is invertible, we can multiply both sides on the left by (Ekt'
to get
1
(E k) EkEk-l ··· E1A
=
1
(Ek) 1
or
Ek-I ··· E1A
=
1
E"k
Next, we multiply both sides by E;!, to get
We continue to multiply by the inverse of the elementary matrix on the left until the
equation becomes
Thus, since the inverse of an elementary matrix is elementary, we have written A as a
product of elementary matrices
...
This is an example of a matrix de
composition
...
We will look at a useful matrix decomposition in the next section and a couple more
of them later in the book
...
[
]
Solution: We row reduce A to I, keeping track of the elementary row operations used:
0 2
1 1
R2 ! Ri
-[
1 1
0 2
]
12R2
-[
1 1
0 1
]
R1 - R1
-[
1 0
0 1
]
Hence, we have
Thus,
and
[o ] [1 OJ [ ]
1
1 1
1
A= e-lE-lE1 2 3 = 1 00 20 1
PROBLEMS 3
...
Multiply each of the elementary matrices by
A =
4
20
and verify that the product EA is
the matrix obtained from A by the elementary row
operation
...
(b) Swap the second and third rows
...
(d) Multiply the second row by 6
...
A2 Write a 4 x 4 elementary matrix that corresponds to
each of the following elementary row operations
...
(b) Swap the second and fourth rows
...
(d) Add
2
A4 For each of the following matrices:
times the first row to the third row
...
(a) A=
it is an elementary matrix and state the correspond
�
ing elementary row operation or explain why it is
not elementary
...
A3 For each of the following matrices, either state that
(c)
sequence
Eb
...
1
(ii) Determine A- by computingEk···E1
...
(a)
a
(d)
en
-[ �0 0 -1�i
�[ ! �I
[� : �]
(b) A=
5
(c) A=
(d) A=
-1 3 -4 -1
0 1 2 0
-2 4 -8 -1
Homework Problems
Bl Write a 3
3 elementary matrix that corresponds
B3 For each of the following matrices, either state that
to each of the following elementary row opera
it is an elementary matrix and state the correspond
tions
...
and verify that the productEA is
-3
the matrix obtained from A by the elementary row
operation
...
(b) Swap the first and third rows
...
(d) Multiply the first row by
(e) Add
(-2)
4 4
2
...
(f) Swap the first and second rows
...
(a) Add 6 times the fourth row to the second row
...
(c) Swap the first and fourth rows
...
(e) Add
(-2)
times the third row to the first row
...
0 1 �0i
(a)[��
[-�0 �1 0�i
[0� 0��i1
(b)
l
o
o
[ ��
0
[�1 0 �1 i
[ l ! �]
�1
(c)
(d)
(e)
cn -
B4 For each of the following matrices:
(i) Find
a
sequence
of
elementary
matrices
Eb
...
(ii) Determine A-1 by computingEk···E1
...
(a) A=
-
(b)
A=
� �
[ �1
H _; -�]
1
(c)
A=
4
3
(d) A=
-2
3
1
-2
4
4
-5
-
2
-1
1
1
2
-3
5
Conceptual Problems
Dl (a) Let L
:
IR
...
2
be the invertible linear op
(a) Determine elementary matrices
£1
and
£2 such
show that L can be written as a composition of
E2E1A= I
...
Instead of using matrix multiplication,
shears, stretches, and reflections
...
erator with standard matrix
A =
[� =�l
By
writing A as a product of elementary matrices,
that
(b)
a composition of shears, stretches, and reflec
E1 b by performing the elemen
tary row operation associated with £1 on the
matrix b
...
forming the elementary row operation associ
(b) Explain how we know that every invertible lin
ear operator L : JRll
D2 For 2
x
-t
IR
...
D3 Let A=
[ � ;]
and
b
=
[;]
...
Solve the system Ax = b by row reducing
[ A I b ]
...