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CHAPTER 4 MATRICES

Matrices

A matrix is a rectangulararray of numbers enclosed in large

brackets

Example:

Matrix A =

1 2 3

4 5 7

Read:Matrix A equals 1, 2, 3,

4, 5, 7

One matrix (singular)

Two or more matrices (plural)

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CHAPTER 4 MATRICES

Forming a Matrix

The following are steps to form a matrix from given information.

Read the information and determine the two groups orcategories

in the information provided.

Draw a table using one group of information across from left to right and

the second group down from top to bottom. Fill in the table with the

numerical data.

Write the numerical data within brackets

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Forming a Matrix

A certain company packs chocolates in packets and boxes. 8 type A and

12 type B chocolates are packed in each packet, while 10 type A and

15 type B chocolates are packed in each box. Form a matrix based on

the given information.

STEP 1

Categories:

Types of chocolates A and B

Types of packing packets and boxes

Example

Solution

STEP 2

Type/packing A B

Packet 8 12

Box 10 15

STEP 3

The matrix formed is

8 12

10 15

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Rows, Columns and Order ofMatrices

The orderof a matrix with m number of rows and nnumber of columns is

given by an expression m x n (read m x n as m by n).

Example

A matrix of order 3 x 2 has 3 rows and 2 columns

1 1

3 9

4 0

Row 1

Row 2

Row 3

Noofrows

NoofColu

mns

3 x 2 matrix read as 3 by 2

Co

lumn1

Co

lumn2

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A row matrix is a matrix with only one row

Example:

3 4 7 5 -6 10

1 x 1 1 x 31 x 2

Only one row

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A column matrix is a matrix with only one column

Example:

3 4

7

5

-6

101 x 1

3 x 1

2 x 1

Only one column

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A square matrix has the same number of rows and columns

Example:

-7 4 5

7 8

5 3 2

-6 0 9

10 13 -41 x 1

3 x 3

2 x 2

No. of rows = No. of columns

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Elements in a Matrix

A 3 x 3 matrix has 3 x 3 = 9 elements. The elements of a matrix refer to the

numbers in the matrix.

Example:

5 3 2

-6 0 9

10 13 -4

3 x 3

5, 3, 2, -6, 0, 9, 10, 13, and -4 are the elements

of the above matrix

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A matrix oforder m x n has mn elements.

A 3 x 4 matrix has 3 x 4 = 12 elements.

Example:

2 4 7 5

6 0 9 1

-3 -5 8 3

3 x 4

2, 4, 7, 5, 6, 0, 9, 1, -3, -5, 8 and 3 are the elements

of the above matrix

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Each element is defined by its position in the matrix. An element in a row i

and columnjin a matrix is represented by aij

Example:2 4 7 5

6 0 9 1

-3 -5 8 3

2 = a11 4 = a12 7 = a13 5 = a14

6 = a21 0 = a22 9 = a23 1 = a24

-3 = a31 -5 = a32 8 = a33 3 = a34

2

2

9

9

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Equal Matrices

Matrices are equal if they have the same number ofrows and the same

number ofcolumns, and if the corresponding elements are equal.

Example:

4 5

7 8A =

4 5

7 8B =

2 x 2 2 x 2

a11 a12

a21 a22

= 4 = b11 = 5 = b12

= 7 = b21 = 8 = b22

A = B

Matrix A and Matrix B are equal since they have the same order and the

corresponding elements aij and bij are equal.

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Equal Matrices

When matrices are equal, elements whose values are unknown can be

determined.

Example:

x + 5 -3

y 4p

7 -3

3 12

=

State the values of the unknowns in the following pairs of equal

matrices.

Solution

x + 5 = 7

x = 7 - 5

x = 2

y = 3

4p = 12

p = 12 4

p = 3

x + 5 7

y 34p 12

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Addition and Subtraction ofMatrices

Matrices can be added and subtracted if they have same order.

Example:

4

6

7

3

=

Addition and subtraction of matrices has the same properties as the

addition and subtraction of numbers.

+

11

9=

4 + 7

6 + 3

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Example:

4

6

7

3

=



-4 - 7

6 - 3

-3

3

=

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Example:

4 -3

-6 5



+7 1

2 9

=

=

4 + 7 -3 + 1

-6 + 2 5 + 9

11 -2

-4 14

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Example:

4 -3

-6 5



-7 1

2 9

=

=

4 -7 -3 - 1

-6 - 2 5 - 9

-3 -4

-8 -4

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Multiplication ofa Matrix by a Number

When a matrix is multiplied by a number, every element in the matrix is

multiplied by the number.

Example:

1 2

3 4

, then 2A =If A =

1 2

3 42A = 2 =

2 x 1 2 x 2

2 x 3 2 x 4

=

2 4

6 8

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Multiplication ofTwo Matrices

Multiplication of two matrices is different from scalar multiplication.

Let us look at the following example.

Table A shows the dinner Ali and Ah Sung had for 4 days.

Table B shows the price of each item.

Table A

Food/

Name

Fried

Rice

Chicken

Rice

Ali 3 1

Ah Sung 2 2

Table B

Food Price

(RM)

Fried rice 4.00

Chickenrice

5.00

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Based on the two tables given, we can find the total expenditure that theyspend on dinner is the sum of the products of the amount and the price

for each food as shown below.

Ali

Ah Sung

Amount x Price Amount x Price+ = Total expenditure3 x 4 +

2 x 4

1 x 5

2 x 5+

=

=

17

18

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The calculation can also be done by matrix method.

Step 1 Write the information in matrix form

Dinner matrix Price matrix

3 1

2 2

and4

5

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Step 2 To find the expenditure, we multiply the matrices as shown below.

Dinner matrix Price matrix

3 1

2 2x

4

5=

3 x 4 + 1 x 5

2 x 4 + 2 x 5

Expenditure matrix

=

12 + 5

8 + 10

=

17

18

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Two matrices can be multiplied if and only if the number ofcolumns of the

first matrix equals the number ofrows of the second matrix.

MATRIX

A

First

matrix

X

Multiplied

by

B

Second

matrix

= C

Order of

product

ORDER 1 x 2 x 2 x 1 = 1 x 1

2 x 1 x 1 x 2 = 2 x 2

2 x 2 x 2 x 2 = 2 x 2

3 x 1 x 1 x 3 = 3 X 3

Example:

No. of columns of first matrix No. of rows of second matrix=

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The product of two matrices of order1x 2and 2 x1 is a matrix of order1x1.

3 4

2

5

=

x3 4 2

5

=

1 x 1

1 x 2

2 x 1

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4.5 Multiplication ofTwo Matrices

1. The product of two matrices of order1x 2and 2 x1 is a matrix of order

1x1.

2. Using the multiplication procedure, multiply the elements of the first row

of the first matrix with the elements of each columns of the second

matrix.

Example:

3 4 2

5

=

= 26

3 x 2 + 4 x 5

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The product of two matrices of order2x 1 and 1 x2is a matrix of order

2x2.

=3

2

1 4x

3

2

1 4 =

2 x 2

2 x 1

1 x 2

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1. The product of two matrices of order2x 1 and 1 x2is a matrix of order

2x2.



matrix.

3. Repeat the step 2 for other rows of the first matrix.

Example:

=2 x 1 2 x 4

3

2

1 43 x 1

=12

2 8

3

3 x 4

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The product of two matrices of order2x 2and 2 x1 is a matrix of order

2x1.

3 1

2 2

4

5 =

3 1

2 2

4

5

=

x

2 x 1

2 x 2

2 x 1

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1. The product of two matrices of order2x 2and 2 x1 is a matrix of order

2x1.



matrix.


Example:

3 1

2 2

4

5

=

=

17

18

3 x 4 + 1 x 5

2 x 4 + 2 x 5

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The product of two matrices of order2x 2and 2 x2is a matrix of order2x2.

=3 1

2 -5

4 0

8 7

=

3 1

2 -5

4 0

8 7x

2 x 2

2 x 2

2 x 2

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1. The product of two matrices of order2x 2and 2 x2is a matrix of order

2x2.



matrix.


Example:

=3 x 0 + 1 x 7

2 x 4 + (-5) x 8 2 x 0 + (-5) x 7

3 1

2 -5

3 x 4 + 1 x 8

=-35-32

720

4 0

8 7

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Example:

=1 p

0 2 43p

q - 17q 4

-3 2

Solving

Matrix Equations

=

43p

q - 17q 3p 4 + 2p

-6 4

-6 = 3p

P = (-6) 3

p = -2

Substitute p = -2 into

q 3p = 7

q 3(-2) = 7

q + 6 = 7

q = 7 6

q = 1

1

22

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Identity Matrices

The product of an identity matrix, I, and any given matrix A of the sameorder as Iis equaltoA.

I x A = A or A x I = A

IA = AI = A

If and only if and A are of the same order.

An identity matrix is usually denoted by Iand is also known as

a unit matrix.

An identity matrix is a square matrix and there is only one identitymatrix for each order. All diagonal elements (from top left to bottom

right) are equal to 1 and the rest are 0.

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4.6 Identity Matrices

Example:

=1 0

0 1 0 x 1 + 1 x 40 x 2 + 1 x (-3)

1 x 1 + 0 x 41 x 2 + 0 x (-3)2 1

-3 4

2 1

-3 4

=

1 0

0 1

is an identity matrix for2 1

-3 4

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Identity Matrices

Example:

=2 1

-3 4 -3 x 0 + 4 x 1-3 x 1 + 4 x 0

2 x 0 + 1 x 12 x 1 + 1 x 01 0

0 1

2 1

-3 4

=

1 0

0 1

is an identity matrix for2 1

-3 4

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Inverse Matrices

IfA is a square matrix, B is another square matrix andA x B = B x A = IthenA is the inverse matrixof matrix B and vice versa.

MatrixA is called the inverse matrixofB for multiplication and vice versa.

The symbolA-1denotes the inverse matrix ofA.

Inverse matrices for multiplication only exist for square matrices but notall square matrices have an inverse matrix for multiplication

IfAB IorBA I, thenA is not the inverse ofB and B is not the inverse

ofA.

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4.7 Inverse Matrices

Example:

Determine whether matrix A =4 1

7 2

is an inverse matrix of matrix of matrix B =2 -1

-7 4

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4.7 Inverse Matrices

Solution :

4 1

7 2

2 -1

-7 4AB =

=

7 x (-1) + 2 x 47 x 2 + 2 x (-7)

4 x (-1) + 1 x 44 x 2 + 1 x (-7)

1 0

0 1

=

AB = I

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Inverse Matrices

Solution :

4 1

7 2

2 -1

-7 4

BA =

=

-7 x 1 + 4 x 2-7 x 4 + 4 x 7

2 x 1 + (-1) x 22 x 4 + (-1) x 7

1 0

0 1

=

BA = IAB = BA = I

A is the inverse matrix of B and vice versa

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Inverse Matrices

a b

c d

If A =

The inverse of matrix can be found using a formula;

, then

A-1 =1 d -b

-c aad - bc

where ad bc 0

ad bc is known as the determinant of matrix A.

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Inverse Matrices

Example 1:

Find the inverse matrix of4 1

7 2

using the formula

Solution :

4 1

7 2

-1

=1

4 x 2 1 x 7 2

4

7

1-

-

1

1 4

2

7

1-

-=

4

2

7

1-

-

=

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Inverse Matrices

Example 2:

Find the inverse matrix of4 1

6 2

using the formula

Solution :

4 1

6 2

-1

=1

4 x 2 1 x 6 2

4

6

1-

-

1

2 4

2

6

1-

-=

2

1

-3

-1

2=

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4.8Solving Simultaneous Linear Equations Using

Matrices

Simultaneous linear equations ax + by = h and cx + dy = k can be writtenin the matrix form as follows;

a b

c d

x

y=

h

k

Example 1: 2x 5y = 7 ; -3x + y = 8 can be written as

2 -5

-3 1=

x

y

7

8

1

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Solving Simultaneous Linear Equations Using

Matrices

Example 2: 5c = 1 ; -4c - d = 5 can be written as

5 0

-4 -1

=

c

d

1

5

Example 3: 5p - q = -4 ; -p + 2q = 0 can be written as

5 -1

-1 2=

p

q

-4

0

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Matrices

Example 2: 5c = 1 ; -4c - d = 5 can be written as

5 0

-4 -1

=

c

d

1

5

Example 3: 5p - q = -4 ; -p + 2q = 0 can be written as

5 -1

-1 2=

p

q

-4

0

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4.8Solving Simultaneous Linear Equations Using

Matrices

a b

c d

x

y=

h

k

Matrix equations in the form

can be solved for unknowns x and y as follows

(a) Leta b

c d

A = , and find A-1.

(b)Multiply both sides of the equation by A-1

a b

c d

x

y=

h

kA-1 A-1

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Matrices

x

y

h

k

(c) 1d -b

-c aad - bc=

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Matrices

Example :Given that3 5

-1 -2

x

y=

2

-7

, find the value of x and y.

Solution : x

y

2

-7

1 -2 -5

1 33 x (-2) 5 x (-1)=

1

-1=

-2 x 2 + (-5) x (-7)

1 x 2 + 3 x (-7)=

-31

19

x = -31, y = 19

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Given that matrix P =

23

56,matrix Q =

63

51 k

m

and PQ =

10

01

(a) Find the value of k and m

(b) Using matrices, calculate the value of x and y thatsatisfy the following matrix equation:

!

7

4

23

56

y

x

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Solution

(a) -1

=

63

51 k

m

23

56

k = -2 m = 6 x (-2) (-5) x 3

= 3

A-1 =1 d -b

-c aad - bcwhere ad bc 0

ad bc is known as the determinant of matrix A.

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Solution

x

y

h

k

1d -b

-c aad - bc

=

!

7

4

23

56

y

x

x

y

4

7

1-2 5

-3 6

3=

1

3=

-2 x 4 +5 x 7

(-3) x 4+6 x 7=

27

30

x = 9, y = 10

1

3

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(a) P1

K1

k = - 2

N1

(b)

N1

K1

3

)53()26(

!

m

!

7

4

63

5-2

3

1

y

x

!

10

9

y

x

x = 9 y = 10

N1

m = x x

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(a) P1

K1

k = - 2

N03

1

)53()26(

!

vv

m

(b)

!

7

4

63

52

3

1

y

x

!

10

9

y

x

N0

K1

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SPM 2003

M is a 2 x 2 matrix where M 3 -25 -4 = 1 00 1

(a) Find the matrix M.

(b) Write the following simultaneous linear equations as

a matrix equation.

3x 2y = 7

5x4y = 9

Hence, calculate the values ofx and y using matrices

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Answer:

3 -2

5 -4

-1

=

1

3 x (-4) (-2) x 5 -4

3

5

-2-

-

1

-2 3

-4

-5

2

=

-3

2

2

5

2

-1=

11 ( a)

SPM 2003

P1

P1

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3-4-2

5

=-1

2

-4

3

2

-5

xy

=7

9

xy

7

9

x = 5

y = 4

P1

K1

N1

N1

SPM 2003

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