Updated DIT4108 Session 4 Matrix.pdf - DIT4108 Applied...

DIT4108 Applied Mathematics for
Science and Technology
Session 4
Basic operations of matrices
Tony Kwan

Agenda
● Introduction
● Properties of Matrices
● Types of Matrices
● Addition and Subtraction
● Multiplication
– Scalar Multiplication
– Matrix Multiplication

Introduction
• What is Matrix?

Introduction
● Why do we use it?

Introduction
● Definitions
– A matrix is an array of numbers
A =
– Denoted with a bold Capital letter
– All matrices have an order (or dimension): The
number of rows  the number of columns.
 A is 2 by 3 or (2  3).
5






23 22 21
13 12 11
a a a
a a a

Introduction
Matrix algebra has at least two advantages:
● Reduces complicated systems of equations to simple
expressions
● Adaptable to systematic method of mathematical
treatment and well suited to computers
Definition:
A matrix is a set or group of numbers arranged in a square or rectangular
array enclosed by two brackets
 
1 1 






 0 3
2 4






d c
b a

PROPERTIES OF MATRICES
7

Properties of Matrices
● A specified number of rows and a specified
number of columns
● Two numbers (rows x columns) describe the
dimensions or size of the matrix.
Examples: 3x3 matrix 2x4 matrix 1x2 matrix











3 3 3
5 1 4
4 2 1







2
3
3
3
0
1
0
1
 
1 1 

Properties of Matrices
● A matrix is denoted by a bold capital letter and the
elements within the matrix are denoted by lower case
letters, e.g. matrix [A] with elements aij














mn ij m m
n ij
in ij
a a a a
a a a a
a a a a
2 1
2 22 21
12 11
...
...
   
i goes from 1 to m
j goes from 1 to n
A
mxn
=
m
A
n

Properties of Matrices
Equality of Matrices
● Two matrices are said to be equal only when all
corresponding elements are equal
● Therefore their size or dimensions are equal as
well










3 2 5
0 1 2
0 0 1










3 2 5
0 1 2
0 0 1
A = B =
A = B

Properties of Matrices
Equality of Matrices
Some properties of equality:
● IIf A = B , then B = A for all A and B
● IIf A = B , and B = C , then A = C for all A , B and C










3 2 5
0 1 2
0 0 1
A =
B =










33 32 31
23 22 21
13 12 11
b b b
b b b
b b b
If A = B then
ij ij
b a 

TYPES OF MATRICES
12

Types of Matrices
1. Column matrix or vector:
The number of rows may be any integer but the number
of columns is always 1










2
4
1






 3
1












1
21
11
m
a
a
a


Types of Matrices
2. Row matrix or vector
● Any number of columns but only one row
 
6 1 1
 
2 5 3 0
 
n
a a a a
1 13 12 11


Types of Matrices
3. Rectangular matrix
Contains more than one element and number of rows is not equal
to the number of columns













6 7
7 7
7 3
1 1






0 3 3 0 2
0 0 1 1 1
n m 

Types of Matrices
4. Square matrix
● The number of rows is equal to the number of columns
(a square matrix A has an order of m)






0 3
1 1


