|
Matrix
& its Elementary Operations
Introduction
Many problems in engineering
and statistics, as well as social science and natural science use matrix
to express relations with factors affecting one and other and to build-up
models to describe possible phenomena. Undoubtedly matrix notations give
clear presentation and help solving of problems.
An m
by n (m x n) matrix
A is a rectangular array arranged in m
rows (horizontal) and n columns (vertical) :
A = 
|
aij
denotes the element in the ith row, jth column of A,
or the i,j entry of A
Example
1 :
A =
A is a 2 x 3 matrix with a12 = 2,
a21 = 4, etc.
Example
2 :
B =
B is a 3 x 1 matrix
Example
3 : C =
C is a 2 x 2 matrix
Example
4 : The following 4 x 4 matrix gives
the distance in km between the indicated cities.
| I.
Scalar Multiplication |
| If A
is a m x n matrix and k is any scalar, then k A
= [ k • aij ] = [ aij
• k ] = A k |
Example
5 : A
=
2 A =  |
| II.
Addition and Subtraction |
| If
A and B are m
x n matrices (i.e., A and B
must be with same dimension) |
| then A
+ B = [ aij +
bij ] |
|
| Subtraction
is similar with addition in the way that A -
B = A + ( -B ) |
|
=
A + (-1)B |
Example
6 : A
= B
=  A
- B = 
| III.
Matrix Multiplication |
| If A
is a m
x matrix
and B is a x k
matrix,
|
| then C
= AB is a m
x k
matrix such that : |
cij =  ais bsj
for any 1 i
m and 1
j n |
| |
Practical
Example
| A
furniture factory receives an order to make enough chairs, book shelves,
desks and tables to furnish a new office building, which contains
26 offices and 7 conference rooms. Each office is to be furnished
with 3 chairs, 2 book shelves, 1 desk and 1 table, whereas each conference
room is to be furnished with 10 chairs, 3 books shelves, 2 desks and
4 tables. It is also known that it takes 4 working hours to make a
chair and costs $900, it takes 2 hours and costs $550 for a book shelf,
3 hours and $800 for a desk, 1 hour and $600 for a table. How many
hours and what is the total cost to make all the furniture ? |
|
| To summarise,
we can tabulate the given data as follow : |
| |
Table
A
|
Office
|
Conference
Room |
|
Building |
26 |
7 |
|
|
Table
C
|
Time(hours)
|
Cost($)
|
| Chairs
|
4 |
900 |
| Book
Shelves |
2 |
550 |
| Desks |
3 |
800 |
| Tables |
1 |
600 |
|
Table B
|
Chairs
|
Book
Shelves |
Desks
|
Tables
|
|
Office |
3 |
2 |
1 |
1 |
|
Conference Room |
10 |
3 |
2 |
4 |
|
To apply matrix notation,
we define : A = [ 26 7 ] ,
B =  , C = 
| That is A
encodes the number of offices and conference room in the building, |
| B
decodes the number of chairs, bookshelves, desks and tables in each
room and |
| C
decodes the number of hours and cost for each item. |
| |
| The matrix product
: AB = [ 26 7 ]
= [ 148 73 40 54 ] |
| gives the number
of chairs, book shelves, desks and table in the building. |
|
|
| The product :
(AB)C = [ 148 73
40 54 ] = [ 912 237750 ]
|
| gives totally
912 hours and $237750 to make all the furniture. |
|
|
 |
| Question
! |
| How
much will it cost and how long
will it take
to make the furniture
for 1 office
& 1 conference room ?
|
|
and C = A + k A
= 
, the value
of k is :
