1
INTRODUCTION TO MATRICES,
SUM and PRODUCT of MATRICES
Lecture 4
1
INTRODUCTION TO MATRICES,
SUM and PRODUCT of MATRICES
Lecture 4
2
SYSTEMS OF LINEAR EQUATIONS
3
SYSTEMS OF LINEAR EQUATIONS
A system of
3 equations with 3 unknowns
x, y, z.
A system of
2 equations with 6 unknowns
x, y, z, u, v, w.
A system of
4 eqautions with 2
unknowns
x, y.
4
x
x
9
x
5
x
5
x
6
x
2
15
x
x
x
4
x
2
x
7
x
4
6
5
4
3
2
1
6
5
4
3
2
1
49
x
7
x
21
0
x
x
8
x
2
x
5
7
x
x
3
2
1
2
1
2
1
2
1
7
x
x
2
x
2
1
x
x
3
x
6
1
x
x
x
2
3
2
1
3
2
1
3
2
1
4
The number of solutions
5
Which mathematical operations
do not
change the solution of a system?
2. We can multiply any equation by some number, e.g. the first eqaution by ‘
3
’:
e.g. for
x + y - 3z
= 4
2x +2y + z
= 6
2x - 7y – 2z
=9
3
x +
3
y -
9
z =
12
2x +2y + z =
6
2x - 7y – 2z = 9
3. We can add equations to one another e.g. the second equation to the third one
x + y - 3z =
4
2x +2y + z =
6
4
x
-
5
y
–
z =
15
2x +2y + z
= 6
2x - 7y – 2z =
9
_______________________________________-___
4x – 5y – z =
15
1. We can swap equations, e.g the first with the third:
2x - 7y – 2z
=9
2x +2y + z
= 6
x + y - 3z
= 4
+
6
- denotes the coefficient standing in the
3
-rd equation by the
4
-th unknown x
4
row column
m
n
mn
2
2
m
1
1
m
2
n
n
2
2
22
1
21
1
n
n
1
2
12
1
11
b
x
a
x
a
x
a
b
x
a
x
a
x
a
b
x
a
x
a
x
a
...
.
..........
..........
..........
..........
...
...
The general form of
m
linear equations in
n
unknowns is
Where the first subscript denotes the row number and the second the column number.
m
Equation
2
Equation
1
Equation
E
E
E
m
2
1
...
..........
.....
a
3
4
a
mn
7
The mentioned operations are called elementary row operations. Generally we can
8
Any other operation which
does not change
the solution is
a combination of the obove ones !!!!
9
Using
elementary row operations
we can transform a system of equations into
an upper triangular form:
x + y + z = 1
2x + 3y +4z = 4
3x + 4y +6z = 5
(-2)E
1
+E
2
x + y + z = 1
y +2z = 2
3x + 4y +6z = 5
(-3)E
1
+E
3
x + y + z = 1
y +2z = 4
y +3z = 2
(-1)E
2
+E
3
x + y + z = 1
y +2z = 2
z = 0
Backward subtitution gives z = 0 , y = 2, x = -1
So the unique solution is (x, y, z) = (-1, 2, 0)
The system has been triangularized
.
10
The Gaussian Eimination Algorithm is a
SYSTEMATIC SEQUENCE OF ELEMENTARY ROW
OPERATIONS
(the most efficient sequence) which allows us to solve the system.
http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/Gauss.html
11
The Gaussian Eimination Algorithm is a
systematic
sequence
of elementary row operations
(the most efficient
sequence).
The process of Gaussian elimination has two parts.
The first part (Forward Elimination) reduces a given system to
either upper triangular form, or results in a degenerate equation
with no solution, indicating the system has no solution. This is
accomplished through the use of elementary operations.
The second step (Backward Elimination) uses back-substitution to
find the solution of the system above.
12
The Gaussian Eimination Algorithm is a systematic
sequence of elementary row operations (the most efficient
sequence).
First
you eliminate the
first unknown
using the
first equation
from all
but the first equation,
Second
you eliminate the
second unknown
using the
second
equation
from all but the first and second equation
Third
you eliminate the
third unknown
using the
third equation
from
all but the first, second and third equation
etc.
You obtain the upper triangular form.
F
o
rw
a
rd
e
li
m
in
a
ti
o
n
B
a
c
k
w
a
rd
E
li
m
in
a
ti
o
n
1.
2.
Next you perform the same procedure starting from the last equation
13
Gaussian Elimination Steps:
1.
Write the augmented matrix for the system of linear
equations.
2. Use elementary row operations on the augmented
matrix [A|b] to transform A into upper triangular
form.
If a zero is located on the diagonal, switch the
rows until a nonzero is in that place. If you are
unable to do so, stop; the system has either infinite
or no solutions
.
3. Use back substitution to find the solution of the
problem.
14
THE MATRIX
15
The coefficient matrix of the above system is
The size of A is
n
m
Where
a
ij
are called the coefficients of the system
b
i
are the constants or right-hand side R-HS.
.
A
mn
m2
m1
2n
22
21
1n
12
11
a
...
a
a
...
...
...
...
a
...
a
a
a
...
a
a
(read: „m by n”).
A matrix
A
with elements
a
ij
is often denoted as [
a
ij
] or
[
a
ij
]
mxn
.
16
The
augmented matrix
A|B of the above system
.
...
...
...
...
...
...
...
...
m
mn
m
m
n
n
b
b
b
a
a
a
a
a
a
a
a
a
B
A
2
1
2
1
2
22
21
1
12
11
17
?
how?
OUR GOAL
18
TYPES OF MATRICES
MORE DEFINITIONS
19
1. A square matrix of order
n
is a matrix of order
n x n
otherwise its a rectangular matrix
Types of matrices:
2. A diagonal matrix of order n is a square matrix of order n which
has all elements lying outside the diagonal equal to 0.
nn
a
a
a
0
0
0
0
0
0
22
11
20
3. The unit (identity) matrix of order n
is a diagonal matrix which has all elements equal to one.
1
0
0
0
1
0
0
0
1
n
I
4. A column matrix is a matrix of order mx1.
5. A vector matrix is a matrix or order 1xn.
6. A lower triangular matrix is a matrix which has
all the elements lying over the main diagonal equal to zero.
7. An upper triangular matrix is a matrix which has
all the elements lying under the main diagonal equal to zero.
nn
n
n
a
a
a
a
a
a
0
0
0
2
22
1
12
11
An identity matrix may be denoted I
n
, I , 1
or E (the latter being an abbreviation
for the German term "Einheitsmatrix„)
..
..
..
..
..
..
21
Definition
Matrices A and B are said to be equal (A = B) if
1. they are of the same size and
2. each element of A is equal to the corresponding element of B:
a
ik
= b
ik
for i = 1, ... , m and k = 1, ... ,n.
8. The zero matrix of order mxn (sometimes called the null matrix)
denoted by 0 or 0
mxn
is a matrix of order mxn which has
all elements equal to zero.
0
0
0
0
0
0
0
0
0
22
]
a
[
A
T
ij
T
,
a
a
ji
T
ij
Definition
Let A = [ a
ij
] be of order m
x
n, then the matrix of order n
x
m,
denoted by
A
T
where
is said to be the transpose of matrix A.
The transpose matrix is obtained by interchanging
rows with
columns.
The transpose of a matrix
1
≤
i
≤
m, 1
≤
j
≤
n,
(A
T
)
T
= A
23
ADDING AND
MULTIPLYING MATRICES
24
Definition
The sum of matrices A = [ a
ij
] and B = [ b
ij
] of
order m n,
is the matrix C = [ c
ij
] of order m n such that
c
ij
= a
ij
+ b
ij
,
1≤ i ≤ m, 1 ≤
j ≤ n.
Definition
The product of a matrix A = [ a
ij
] and a real number c
R
is the matrix:
cA =[ c a
ij
].
To multiply a matrix A by a number c, we multiply every
element of the matrix by the number c.
25
1
z
y
x
c
b
a
1
cz
by
ax
In 'equation language'
First we shall multiply two matrices:one row matrix
R
and one column matrix
C
n
2
1
n
2
1
c
c
c
C
and
r
r
r
R
)
,
,
,
(
THE PRODUCT
26
Example
The product of 2x2 matrices
27
8
7
8
7
:
4
6
5
6
5
:
3
4
3
4
3
:
2
2
1
2
1
:
1
4
4
22
21
12
11
4
22
4
21
4
12
4
11
3
3
22
21
12
11
3
22
3
21
3
12
3
11
2
2
22
21
12
11
2
22
2
21
2
12
2
11
1
1
22
21
12
11
1
22
1
21
1
12
1
11
y
x
a
a
a
a
y
a
x
a
y
a
x
a
Equation
y
x
a
a
a
a
y
a
x
a
y
a
x
a
Equation
y
x
a
a
a
a
y
a
x
a
y
a
x
a
Equation
y
x
a
a
a
a
y
a
x
a
y
a
x
a
Equation
8
6
4
2
7
5
3
1
:
4
3
2
1
4
3
2
1
22
21
12
11
y
y
y
y
x
x
x
x
a
a
a
a
Equations
All
Application
28
,
b
a
...
b
a
b
a
c
pj
ip
j
2
2
i
j
1
1
i
ij
.
b
a
]
c
[
B
A
n
m
p
1
k
kj
ik
mxn
ij
Definition
Let matrix A = [ a
ij
] be of order m
x
p and matrix B = [ b
jk
] be of order p
x
n.
The product of matrices
A
and
B
is the matrix C= [ c
ij
] of order m
x
n :
where 1
≤
i
≤
m, 1
≤
j
≤
n,
Product of two matrices of arbitrary size
29
In case A and B
fail to 'match
' i.e. A is m
x
p and B is q
x
n and
p ≠ q then no product is defined.
CAUTION
30
CAUTION !!!
BA
AB
2
3
1
2
3
2
1
1
8
5
3
1
A
B
,
3
1
5
4
B
A
A=
B =
a · b = a · c a, b are Real
Numbers
b = c
Also the cancellation law fails for matrices i.e
31
CAUTION
0
B
A
but neither
A
= 0 or
B
= 0
1
1
1
1
B
1
1
1
1
A
2
2
2
2
A
B
Note that
32
For matrices
A
and
B
we obtain:
( A + B )
T
= A
T
+ B
T
;
( A B )
T
= B
T
A
T
.
33
The properties of matrix operations
The operations of sums and products have the
following properties:
•A + B = B + A (commutative law);
•(A + B) + C = A + (B + C) (associative law);
•A + 0 = 0 + A where 0 is the zero matrix;
•A
(B
C) = (A
B)
C (associative law);
•A
(B + C) = A
B + A
C (first distributive
law);
•(A + B)
C = A
C + B
C (second distributive
law);
• Let A = [ a
ij
]
nxn
and I
n
be a unit matrix of order
n,
then A
I
n
= A = I
n
A.
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7
3
8
3
9
5
6
8
4
3
1
2
A
A submatrix of a given matrix
A
is an array obtained by deleting any combintaion
of rows and columns from
A
.
is a submatrix of
A
because
B
is a result of deleting the second row and the second
and third columns of
A
.
DEFINITION
For example,
let
then
7
3
4
2
B
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Time to finish