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More About This Textbook
Overview
Essential Mathematics provides a single unified course of study which addresses all the key skills outcomes.
Product Details
Read an Excerpt
Cambridge University Press
0521612527  Essential  Advanced General Mathematics  by Michael Evans, Sue Avery, Kay Lipson and Doug Wallace
Excerpt
C H A P T E R
1
Matrices
Objectives
▆ To be able to identify when two matrices are equal
▆ To be able to add and subtract matrices of the same dimensions
▆ To be able to perform multiplication of a matrix and a scalar
▆ To be able to identify when the multiplication of two given matrices is possible
▆ To be able to perform multiplication on two suitable matrices
▆ To be able to find the inverse of a 2 × 2 matrix
▆ To be able to find the determinant of a matrix
▆ To be able to solve linear simultaneous equations in two unknowns using an inverse matrix
1.1 Introduction to matrices
A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix.
The following are examples of matrices:
Matrices vary in size. The size, or dimension, of the matrix is described by specifying the number of rows (horizontal lines) and columns (vertical lines) that occur in the matrix.
The dimensions of the above matrices are, in order:
The first number represents the number of rows and the second, the number of columns.
Example 1
Write down the dimensions of the following matrices.
The use of matrices to store information is demonstrated by the following two examples.
Four exporters A, B, C and D sell televisions (t), CD players (c), refrigerators (r) and washing machines (w). The sales in a particular month can be represented by a 4 × 4 array of numbers. This array of numbers is called a matrix.
From the matrix it can be seen:
Exporter A sold 120 refrigerators, 95 CD players, 370 washing machines and 250 televisions.
Exporter B sold 430 refrigerators, 380 CD players, 1000 washing machines and 900 televisions.
The entries for the sales of refrigerators are made in column 1.
The entries for the sales of exporter A are made in row 1.
The diagram on the right represents a section of a road map. The number of direct connecting roads between towns can be represented in matrix form.
If A is a matrix, a_{ij} will be used to denote the entry that occurs in row i and column j of A. Thus a 3 × 4 matrix may be written
For B, an m × n matrix
Matrices provide a format for the storage of data. In this form the data is easily operated on. Some graphics calculators have a builtin facility to operate on matrices and there are computer packages which allow the manipulation of data in matrix form.
A car dealer sells three models of a certain make and his business operates through two showrooms. Each month he summarises the number of each model sold by a sales matrix S:
So, for example, s_{12} is the number of sales made by showroom 1, of model 2.
If in January, showroom 1 sold three, six and two cars of models 1, 2 and 3 respectively, and showroom 2 sold four, two and one car(s) of models 1, 2 and 3 (in that order), the sales matrix for January would be:
A matrix is, then, a way of recording a set of numbers, arranged in a particular way. As in Cartesian coordinates, the order of the numbers is significant, so that although the matrices
have the same numbers and the same number of elements, they are different matrices (just as (2, 1), (1, 2) are coordinates of different points).
Two matrices A, B, are equal, and can be written as A = B when
▆ each has the same number of rows and the same number of columns
▆ they have the same number or element at corresponding positions.
Example 2
If matrices A and B are equal, find the values of x and y.
Although a matrix is made from a set of numbers, it is important to think of a matrix as a single entity, somewhat like a 'super number'.
Example 3
There are four rows of seats of three seats each in a minibus. If 0 is used to indicate a seat is vacant and 1 is used to indicate a seat is occupied, write down a matrix that represents
a the 1st and 3rd rows are occupied but the 2nd and 4th rows are vacant
b only the seat on the front left corner of the bus is occupied.
Example 4
There are four clubs in a local football league.
Team A has 2 senior teams and 3 junior teams
Team B has 2 senior teams and 4 junior teams
Team C has 1 senior team and 2 junior teams
Team D has 3 senior teams and 3 junior teams
Represent this information in a matrix.
Note: rows represent teams A, B, C, D and columns represent the number of senior and junior teams respectively.
Exercise 1A
1 Write down the dimensions of the following matrices.
2 There are 25 seats arranged in five rows and five columns. If 0, 1 respectively are used to indicate whether a seat is vacant or occupied, write down a matrix which represents the situation when
a only seats on the two diagonals are occupied
b all seats are occupied.
3 If seating arrangements (as in 2) are represented by matrices, consider the matrix in which the i, j element is 1 if i = j, but 0 if i ≠ j. What seating arrangement does this matrix represent?
4 At a certain school there are 200 girls and 110 boys in Year 7, 180 girls and 117 boys in Year 8, 135 and 98 respectively in Year 9, 110 and 89 in Year 10, 56 and 53 in Year 11 and 28 and 33 in Year 12. Summarise this information in matrix form.
5 From the following, select those pairs of matrices which could be equal, and write down the values of x, y which would make them equal.
6 In each of the following find the values of the pronumerals so that matrices A and B are equal.
7 A section of a road map connecting towns A, B, C and D is shown. Construct the 4 × 4 matrix which shows the number of connecting roads between each pair of towns.
8 The statistics for the five members of a basketball team are recorded as follows.
Player A: points 21, rebounds 5, assists 5
Player B: points 8, rebounds 2, assists 3
Player C: points 4, rebounds 1, assists 1
Player D: points 14, rebounds 8, assists 60
Player E: points 0, rebounds 1, assists 2
Express this data in a 5 × 3 matrix.
1.2 Addition, subtraction and multiplication by a scalar
Addition will be defined for two matrices only when they have the same number of rows and the same number of columns. In this case the sum of two matrices is found by adding corresponding elements. For example,
Subtraction is defined in a similar way. When the two matrices have the same number of rows and the same number of columns the difference is found by subtracting corresponding elements.
Example 5
It is useful to define multiplication of a matrix by a real number. If A is an m × n matrix, and k is a real number, then kA is an m × n matrix whose elements are k times the corresponding elements of A. Thus
These definitions have the helpful consequence that if a matrix is added to itself, the result is twice the matrix, i.e. A + A = 2A. Similarly the sum of n matrices each equal to A is n A (where n is a natural number).
The m × n matrix with all elements equal to zero is called the zero matrix.
Example 6
Example 7
Using a graphics calculator
A graphics calculator can be used to perform operations with matrices. Entering the required matrices into a TI83/Plus or TI84 + graphics calculator is done as follows.
Select □ EDIT, 1: [A] □ to enter details for matrix A.
Once the desired dimension of the matrix is entered, complete the details by entering the required values for each element. Once the details are entered, return to home screen (□) and repeat to enter details for other matrices.
Once the required matrices have been entered, any operation can be performed on them by selecting the required matrices and using the usual arithmetic operators.
For example to find A + B, select □ 1:[A] □, and then enter □ 1:[B] □.
It should also be noted that any decimal answers using the graphics calculator can be converted to fraction form in the usual way.
Using a CAS calculator
Matrices are accessed through the Matrix Editor. Press the □ key and select Data/Matrix Editor and then 3:New.
From the resulting menu select 2:Matrix. Call this first matrix a and define it as a 2 × 2 matrix.
Press □ to obtain the edit screen. Note that the status in the top left of the screen is now Mat 2 □ 2. The entries are made in the usual way. This is defined as matrix a. The matrix b = □ is defined in a similar way. Return to the home screen.
The two matrices can be viewed by entering a and then b in the entry line.
Addition and multiplication can be undertaken as shown in the screens opposite. Scalar multiplication on a can be performed by entering ka in the entry line.
Exercise 1B
Find X + Y, 2X, 4Y + X, X  Y, 3A and 3A + B.
2 Each showroom of a car dealer sells exactly twice as many cars of each model in February as in January. (See example in section 1.1.)
Find 2A, 3A and 6A.
4 A, B, C are m × n matrices. Is it true that
8 Matrices X and Y show the production of four models a, b, c, d at two automobile factories P, Q in successive weeks.
Find X + Y and write what this sum represents.
1.3 Multiplication of matrices
Multiplication of a matrix by a real number has been discussed in the previous section. The definition for multiplication of matrices is less natural. The procedure for multiplying two 2 × 2 matrices is shown first.
Note that AB ≠ BA.
If A is an m × n matrix and B is an n × r matrix, then the product AB is the m × r matrix whose entries are determined as follows.
To find the entry in row i and column j of AB single out row i in matrix A and column j in matrix B. Multiply the corresponding entries from the row and column and then add up the resulting products.
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