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MATRICES:Matrix Representation, ADDITION AND SUBTRACTION OF MATRICES

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MTH001 ­ Elementary Mathematics
LECTURE # 20
MATRICES
OBJECTIVES
The objectives of the lecture are to learn about:
·
Matrices
EXAMPLE 1
An athletic clothing company manufactures T-shirts and sweat shirts in four
differents sizes, small, medium, large, and x-large. The company supplies two
major universities, the U of R and the U of S. The tables below show
September's clothing order for each university
University of S's September Clothing Order
S
M
L
XL
T-shirts
100
300
500
300
sweat shirts
150
400
450
250
University of R's September Clothing Order.
S
M
L
XL
T-shirts
60
250
400
250
sweat shirts
100
200
350
200
Matrix Representation
The above information can be given by two matrices S and R as shown
below.
S=
R=
MATRIX OPERATIONS
The matrix operations can be summarized as under:
·
Organize and interpret data using matrices
·
Use matrices in business applications
·
Add and subtract two matrices
·
Multiply a matrix by a scalar
·
Multiply two matrices
·
Interpret the meaning of the elements within a product matrix
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MTH001 ­ Elementary Mathematics
PRODUCTION
The clothing company production in preparation for the universities'
September orders is shown by the table and corresponding matrix P below.
S
M
L
XL
T-shirts
300
700
900
500
sweat shirts
300
700
900
500
P=
ADDITION AND SUBTRACTION OF MATRICES
The sum or difference of two matrices is calculated by adding or subtracting the
corresponding elements of the matrices.
To add or subtract matrices, they must have the same dimensions.
PRODUCTION REQUIREMENT
Since the U of S ordered 100 small T-shirts and the U of R ordered 60, then
althogether 160 small T-shirts are required to supply both universities. Thus, to
calculate the total number of T-shirts and sweat shirts required to supply both
universities, add the corresponding elements of the two order matrices as shown
below.
+
=
OVERPRODUCTION
Since the company produced 300 small T-shirts and the received orders for only
160 small T-shirts, then the company produced 140 small T-shirts too many.
Thus, to determine the company's over-production, subtract the corresponding
elements of the total order matrix from the production matrix as shown below.
-
=
MULTIPLY A MATRIX BY A SCALAR
Given a matrix A and a number c, the scalar multiplication cA is computed by multiplying the scalar
c by every element of A .For example:
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MTH001 ­ Elementary Mathematics
MULTIPLICATION OF MATRICES
To understand the reasoning behind the definition of matrix multiplication,
let us consider the following example.
Competing Companies, A and B, sell juice in 591 mL, 1 L and 2 L plastic bottles
at prices of Rs.1.60, Rs.2.30 and Rs.3.10, respectively. The table below
summarises the sales for the two companies during the month of July.
591mL
1L
2L
Company A
20,000
5,500
10,600
Company B
18,250
7,000
11,000
What is total revenue of Company A?
What is total revenue of Company B?
Matrices may be used to illustrate the above information.
As shown at the right, the sales can be written as
a 2X3 matrix, S, the selling prices can be written as a column matrix, P, and the
total revenue for each company can be expressed as a column matrix, R.
S=
R=
P=
Since revenue is calculated by multiplying the number of sales by the
selling price, the total revenue for each company is found by taking the
product of the sales matrix and the price matrix.
Consider how the first row of matrix S and the single column P lead to the
first entry of R.
With the above in mind, we define the product of a row and a column to be the
number obtained by multiplying corresponding entries (first by first, second by
second, and so on) and adding the results.
MULTIPLICATION RULES
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MTH001 ­ Elementary Mathematics
If matrix A is a m n matrix and matrix B is a n p matrix, then the product AB is
the m p matrix whose entry in the i-th row and the j-th column is the product of
the i-th row of matrix A and the j-th row of matrix B.
The product of a row and a column is the number obtained by multiplying
corresponding elements (first by first, second by second, and so on).
To multiply matrices, the number of columns of A must equal the number of rows
of B.
MULTIPLICATION RULES For example
Given the matrices below, decide if the indicated product exists. And, if the
product exists, determine the dimensions of the product matrix.
MULTIPLICATION CHECKS
The table below gives a summary whether it is possible to multiply two matrices.
It may be noticed that the product of matrix A and matrix B is possible as the
number of columns of A are equal to the number of rows of B. The product BA is
not possible as the number of columns of b are not equal to rows of A.
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MTH001 ­ Elementary Mathematics
Does a product exist?
Dimensions of  (Is it possible to multiply Dimensions of
Product
the Matrices
the given
Product Matrix
matrices in this order?)
Yes, the product exists
A=3 3  B=3 2
since the
inner dimensions match
32
AB
(# of columns of A = #
of rows of B).
No, the product does
B=3 2
A = 3×3
not exist
since the inner
dimensions do
n/a
BA
not match
(# of columns of B  #
of rows of A).
MULTIPLICATIVE INVERSES
Real Numbers
Two non-zero real numbers are multiplicative inverses of each other if their
products, in both orders, is 1. Thus,
the multiplicative inverse of a real number, x is
or
since x
= 1 and
x
= 1.
Example:
The multiplicative inverse of 5 is
since
5
= 1 and
5=1
Matrices
Two 2 2 matrices are inverses of each other if their products, in both orders, is
a 2 2 identity matrix. Thus, the multiplicative inverse of a 2 2 matrix, A is
since A
=
and
A=
Example:
3  -1
The multiplicative inverse of a matrix,
is
since is
-5  2
=
=
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MTH001 ­ Elementary Mathematics
-1⎤  ⎡ 1 0
2
1⎤ ⎡ 3
⎥ = ⎢ 0 1
5
3⎥ ⎢-5
2⎦  ⎣
⎦⎣
-1⎤ ⎡2
3
1⎤  ⎡ 1 0
=
⎢-5
2 ⎥ ⎢5
3⎥  ⎢ 0 1
⎦⎣
⎦⎣
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Table of Contents:
  1. Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION
  2. Truth Tables for:DE MORGAN’S LAWS, TAUTOLOGY
  3. APPLYING LAWS OF LOGIC:TRANSLATING ENGLISH SENTENCES TO SYMBOLS
  4. BICONDITIONAL:LOGICAL EQUIVALENCE INVOLVING BICONDITIONAL
  5. BICONDITIONAL:ARGUMENT, VALID AND INVALID ARGUMENT
  6. BICONDITIONAL:TABULAR FORM, SUBSET, EQUAL SETS
  7. BICONDITIONAL:UNION, VENN DIAGRAM FOR UNION
  8. ORDERED PAIR:BINARY RELATION, BINARY RELATION
  9. REFLEXIVE RELATION:SYMMETRIC RELATION, TRANSITIVE RELATION
  10. REFLEXIVE RELATION:IRREFLEXIVE RELATION, ANTISYMMETRIC RELATION
  11. RELATIONS AND FUNCTIONS:FUNCTIONS AND NONFUNCTIONS
  12. INJECTIVE FUNCTION or ONE-TO-ONE FUNCTION:FUNCTION NOT ONTO
  13. SEQUENCE:ARITHMETIC SEQUENCE, GEOMETRIC SEQUENCE:
  14. SERIES:SUMMATION NOTATION, COMPUTING SUMMATIONS:
  15. Applications of Basic Mathematics Part 1:BASIC ARITHMETIC OPERATIONS
  16. Applications of Basic Mathematics Part 4:PERCENTAGE CHANGE
  17. Applications of Basic Mathematics Part 5:DECREASE IN RATE
  18. Applications of Basic Mathematics:NOTATIONS, ACCUMULATED VALUE
  19. Matrix and its dimension Types of matrix:TYPICAL APPLICATIONS
  20. MATRICES:Matrix Representation, ADDITION AND SUBTRACTION OF MATRICES
  21. RATIO AND PROPORTION MERCHANDISING:Punch recipe, PROPORTION
  22. WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS
  23. WHAT IS STATISTICS?:COMPONENT BAR CHAR, MULTIPLE BAR CHART
  24. WHAT IS STATISTICS?:DESIRABLE PROPERTIES OF THE MODE, THE ARITHMETIC MEAN
  25. Median in Case of a Frequency Distribution of a Continuous Variable
  26. GEOMETRIC MEAN:HARMONIC MEAN, MID-QUARTILE RANGE
  27. GEOMETRIC MEAN:Number of Pupils, QUARTILE DEVIATION:
  28. GEOMETRIC MEAN:MEAN DEVIATION FOR GROUPED DATA
  29. COUNTING RULES:RULE OF PERMUTATION, RULE OF COMBINATION
  30. Definitions of Probability:MUTUALLY EXCLUSIVE EVENTS, Venn Diagram
  31. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:ADDITION LAW
  32. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:INDEPENDENT EVENTS