Matrices

Given below is a record of all the types of books kept in a class library.

Type Fiction General

Malay 25 47

Chinese 40 72

English 80 85

The numbers in the table can be presented in a rectangular array.

The first column represents the number of books which are fiction and the first row represents the number of books in Malay, and so on.

8580

7240

4725

• A set of numbers arranged in this manner is called a matrix ( plural – matrices)

• The numbers within each matrix are the elements.

• In the above matrix, there are 3 rows and 2 columns.

• This matrix has a order of 3 by 2 or it is a

3 x 2 matrix.

• In general an m x n matrix refers to m rows and n columns. Its order is m x n

• Column matrix : If the matrix only has one singular column

• Row matrix : If the matrix only has one singular row.

•

80

40

25

804725

• Square matrix : Is when the matrix has an equal number of rows & columns.

• If every elements in a matrix is zero then its is a null matrix ( zero matrix) ; It is denoted by 0

6525

8045

00

00

• Identity matrix

Given that matrix A has an inverse, which is A-1, Then when matrix A is multiplied with the inverse matrix A-1 , it will give a matrix called identity matrix.

A A-1 = I

10

01

100

010

001

• Singular Matrix.

Given that the determinant of A is 0.

i.e. det A = 0, then the inverse of A does not exist.

When a matrix does not have an inverse then that

matrix is called the Singular matrix.

For example this matrix does not have an inverse,

it is a singular matrix.

63

84

When are 2 matrices equal??

If 2 matrices A and B

– are of the same order

– have corresponding elements that are equal

Then, we can say that they are equal.

Addition & Subtractionof Matrices

Given that P & Q are matrices of the same

order

P + Q =

=

1 2 1 2

3 4 3 4

p p q q

p p q q

1 1 2 2

3 3 4 4

p q p q

p q p q

P - Q =

=

Note:

If the order of P and Q are not the same, their sum and

difference are not defined

1 2 1 2

3 4 3 4

p p q q

p p q q

1 1 2 2

3 3 4 4

p q p q

p q p q

Scalar Multiplication

• Multiplication of a matrix by a real number

kP =

where k is a real number

1 2

3 4

kp kp

kp kp

Matrix Multiplication

• Possible only if

no. of columns in left matrix = no. of rows in right matrix

• Multiplying m x n matrix by n x p matrix will result in an m x p matrix

ie:

A 3x3 matrix multiplied by a 3x2 matrix result in a 3x2 matrix.

A 3x2 matrix multiplied by a 2x3 matrix result in a 3x3 matrix.

A 3x2 matrix multiplied by a 3x3 matrix is not posible

To multiply 2 matrices:

=

=

3 2 2 1 0

1 7 4 6 3

1 2

(3 2) (2 4) (3 1) (2 6) (3 0) (2 3)

(1 2) (7 4) (1 1) (7 6) (1 0) (7 3)

1 2

14 15 16

30 43 21

2 x 2 2 x 3

2 x 3

Some Properties of Matrix Multiplication

• NOT commutative

ie: AB ≠ BA

• Is associative

ie: ABC = (AB)C = A(BC)

Commutative and associative properties

surprised to learn that matrix multiplication is not

commutative but associative.

Explain using a problem

35465210521510IVIIIII I

DalePlainHill

203153202122

WindowsDoors

IVIIIIII

Commutative and associative properties

35465210521510IVIIIII I

DalePlainHill

203153202122

WindowsDoors

IVIIIIII

80100

Prices

A CB

Calculate (AB)C and A(BC). Explain why they are equal.

Page 3Page 3

What information does the matrix AB gives? BA?

(AB)C is calculated by first finding the number of doors and windows in each of the developments and then finding the total cost of windows and doors for each development.

A(BC) is calculated by first finding the cost of the doors and windows in each model and then finding their total cost for each development.

*Determinant of n n matrixGeneral Formula for the Determinant

Let A be a square matrix of order n. Write A = (aij), where aij is the entry

on the row number i and the column number j, for i =1, …, n and j = 1,, n.

We have

for any fixed i, and

for any fixed j

where Aij (called the cofactors) is the determinant of the square

matrix of order (n-1) obtained from A by removing the row number i and the column number j multiplied by (-1)i+j.

Inverse of a matrix

Definition of inverse

An n x n matrix A is said to be invertible if we can find an n x n matrix B such that

AB = BA = I

B is called the inverse of A and is usually denoted A-1.

Inverse of a matrix

The inverse of a square matrix does not always exist.

If the inverse exists, the matrix is defined to be non-singular (i.e. det A 0)

If the inverse does not exist the matrix is singular(i.e. det A =0).

Inverse of a matrixThe importance of the inverse matrix can be seen from the solution of a set of algebraic linear equations such as

Ax = b

If the inverse A-1 exists then pre-multiplying both sides gives

A-1Ax = A-1b

Ix = A-1b

and since pre-multiplying a column vector of length n by the nth order identity matrix I does not affect its value, this process gives an explicit solution to the set of equations

x = A-1b

Using matrices to solve Simultaneous Equations

• We can solve simple simultaneous equations like x +3y = 6 and 2x + y = 4 by using matrices.

• It can be represented by matrix equation

x + 3y 6

2x + y 4

or 1 3 x 6

2 1 y 4

How to solve simultaneous equations using matrices?

• This is in the form of AX = B

where A 1 3 X x and B 6

2 1 y 4

If A has an inverse ( ie A-1)

Then A-1 A X = A-1 B

ie I X = A-1 B

X = A-1 B

To find the variables x and y• First, determine if determinant A exists

det A = ( 1 x 1) – (2 x 3)

= 1 – 6 = -5

A-1 1 1 -3

-5 -2 1

Hence, X = A-1 B 1 1 -3 6

-5 -2 1 4

2 3

5 4

• Thus, x 6 y 8

5 5

To check, subsitute x = 6/5 and y =8/5 back into the original set of equations.

Other Applications

• We are given two matrices X and Y that show the price of 3 brands of cat food sold at two stores, for months May and June respectively.

• X 9.4 9.6 Y 9.8 10.0

10.2 10.4 10.2 10.2

11.4 11.0 12.0 12.0

ai) Evaluate Y - X• First, determine if the operation can take

place. This means that the order of the two matrices are the same.

Y – X = 9.8 - 9.4 10.0 - 9.6

10.2 - 10.2 10.4 - 10.2

12.0 - 11.4 12.0 - 11.0

= 0.4 0.4

0 -0.2

0.6 1.0

Explain what the numbers in the answer represent.

• The values in the matrix shows the increase or decrease in prices of the three brands of cat food by comparing the June prices against the May prices.

b) Write down a matrix M such that XM will show the total costs for

each brand of cat food

• First, decide which matrix to use. In this case, its matrix X. Write down the order of the matrix.

• Thus, matrix X is a 3 x 2 matrix.

Know order of final matrix

• Also, there is a need to know the order of the final matrix. In this case, its going to be a 3 x 1 matrix to show the total cost for each of the brands.

• X M = Total cost for

9.4 9.6 brand A

10.2 10.4 brand B

11.4 11.0 brand C

To find the order of matrix M

• X M = Total cost

• ( 3 x 2) M = ( 3 x 1 )

• Thus M needs to be a (2 x 1) matrix.

To find total cost, multiply the two matrices together

Total cost 9.4 9.6 50

In cents = 10.2 10.4 50

11.4 11.0

950

= 1030

1120

What are the values of m1, m2, m3?

• The amount spend in May and June for Cat food on Brand A = 950 cents = $9.50 Brand B = $10.30 and Brand C = $ 11.20

• The prices are given in 50 cents per 100g.

• If the Question asks for 10 tins of 500g, thus we need to multiply the cost 50 cents by 50.

Matrices

Passes At Annual Output

PSLE1 GCE O Level2 GCE A Level3 ITE4 Polytechnic5 University6Year

Per Cent Number

1995 94.2 90.9 86.3 7325 11008 7926

1996 95.5 90.8 85.4 5581 12105 8218

1997 95.7 90.3 84.2 4918 12919 8679

1998 95.0 91.1 86.0 6234 13904 9331

1999 96.2 92.1 86.5 8501 14641 9463

2000 95.8 92.3 85.4 8427 15074 9406

Storing and organising numerical data

6 rows and 7 columns of data, we say the matrix has size (order) 6 7

Page 1Page 1

Operations with Matrices Page 2Page 2

(a) Form a matrix of prices and use it to find the total amount taken on each of the 3 days.(b) What information would be found by pre-multiplying V by (1 1 1)?

7572659836783611595

ChocolateCoffeeTea

V

The prices of the drinks per cup are 55¢ for tea, 60¢ for coffee and 75¢ for drinking chocolate.

explain why the matrix of prices can be a 31 or

13 matrix in (a) and what information is found if

V is post-multiplied by

111

Routes Matrices Page 4Page 4

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D gives only 1 route because it is one-way only.

0111011101210011002110210

R =