Nota Math Matrik

Mathematics QM 016

Topic 5: Matrices And System Of Linear Equations-Lesson Plan

TOPIC : 5.0 MATRICES AND SYSTEMS OF LINEAR EQUATIONS

SUBTOPIC : 5.1 Matrices

LEARNING

OUTCOMES :

(a) Define a matrix and the equality of matrices

(b) Identify the different types of matrices such as row, column,

zero, diagonal, upper triangular, lower triangular and identity

matrices.

(c) Perform operations on matrices such as addition, subtraction,

scalar multiplication and multiplication of two matrices.

The result Euro 2006 ( Group B )

Team GP W D L Pts

France 3 2 1 0 7

England 3 2 0 1 6

Croatia 3 0 2 1 2

Switzerland 3 0 1 2 2

The above standing shows MATRIX form.

What is rows and what is columns.??

Definition

A matrix is a rectangular array of numbers enclosed between brackets.

The general form of a matrix with m rows and n columns is

mnmmm

n

n

n

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211

m rows

n columns

The order or dimension of a matrix of m rows and n columns is m x n.

The individual numbers that makes up a matrix are called its entries or elements,

ija and they are specified by their row and column position.

The matrix for which the entry is in thi row and thj column is denoted by [ ija ].

Mathematics QM 016


Example

Let

732

6521

A

(a) What is the order of matrix A ?

(b) If A=[ ija ], identify 21a and 13a .

Solution

(a) Since A has 2 rows and 3 columns, the order of A is 2 x 3.

(b) The entry 21a is in the second row and the first column. Thus, 21a = -2.

The entry 13a is in the first row and the third column, and so 13a = 21 .

Example 1

Given 3 3ij x

A a

Find matrix A if 2

ij

ij, i ja

j i, i j

Equality of Matrices

Two matrices are equal if they have the same dimension and their corresponding entries

are equal.

Example 2

Which matrices below are the same ?

12

21A , 21B , 12C ,

12

21D ,

12

12E ,

12

12

21

F

Solution

A = D

Example 3

Let

248

463

b

aA , and

2832

469

d

cB .

If A = B, find value of a, b, c, and d.

Mathematics QM 016


Types of Matrices

Row Matrix is a (1 x n) matrix ( one row );

naaaaA 1131211

Example

54321A ; 5348701B

Column Matrix is a (m x 1) matrix ( one column );

1

31

21

11

ma

a

a

a

A

Example

0

4A

,

2

3

5

7

B

Square Matrix is a nxn matrix which has the same number of rows as columns.

Example

81

31A , 2 x 2 matrix.

132

213

231

B , 3 x 3 matrix.

Zero Matrix is a (m x n) matrix which every entry is zero, and denoted by O.

Example

0 0 0

0 0 0

0 0 0

O

,

0 0

0 0

0 0

O

, 0 0

0 0O

Mathematics QM 016


Diagonal Matrix

Let A =

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

m

m

m

m m m mm

a a a a

a a a a

a a a a

a a a a

The diagonal entries of A are 11 22 33a , a , a ,…, mma

A square matrix which non-diagonal entries are all zero is called a diagonal

matrix.

Example

(a) 2 0

0 3A

(b)

1 0 0

0 2 0

0 0 3

B

(c)

0 0

0 0 0

0 0

a

C

b

Identity Matrix is a diagonal matrix in which all its diagonal entries are 1, and

denoted by I.

Example

(a) 2 2

1 0

0 1A I

(b) 3 3

1 0 0

0 1 0

0 0 1

B I

Lower Triangular Matrix is a square matrix and 0ija for i j

11 12 12

21 22 23

31 32 33

a a a

A a a a

a a a

Example

A =

323

023

001

B =

edc

fb

a

0

00

Mathematics QM 016


Upper Triangular Matrix is a square matrix and 0ija for i j

11 12 12

21 22 23

31 32 33

a a a

A a a a

a a a

Example

A =

300

420

321

B =

f

ed

cba

00

0

Operations on Matrices

Addition And Subtraction Of Matrices

For m x n matrices, A = [ ija ] and B = [ ijb ],

A + B = C = ij mxnc , where ijijij bac .

A – B = D = ij mxnd , where .ijijij bad

Note

The addition or subtraction of two matrices with different orders is not defined. We say

the two matrices are incompatible.

Example 4

Simplify the given quantity for

43

21A ,

65

34B and

2

1C .

(a) A + B (b) A – B (c) A + C

Scalar Multiplication

If c is a scalar and ijA a then ijcA b where ij ijb ca

Mathematics QM 016


Example 5

Given

2 4

8 5

6 7

A

, find 1

2A .

Example 6

Let 1 4

5 3A

and 3 6

4 2B

. Calculate 3 2A B.

Properties

(a) A B B A ( Commutative )

(b) A B C A B C ( Associative)

(c) A A A A O ( O- zero matrix)

(d) A A A , constant

(e) A B A B

(f) A A

Exercises

1. Identify the order of the given matrix.

(a)

654

321 (b)

10

01

01

(c)

d

c

b

a

(d) kji

Mathematics QM 016


2. (a) Find matrix 2 3ij x

A a if 2 2ija i j j i

(b) Find matrix 3 3ij x

B b if

2

2

ij

i j, i j

b ij, i j

i j, i j

3. Simplify the given quantity for

53

21A and

94

12B .

(a) A + B (b) A – B

(c) 2A – 5B (d) 3A + 2B

4. Solve the given equation for the unknown matrix X.

(a) 2X +

456

321 =

000

000

(b) -2

42

004

33

62XX

Answers

1. (a) 2 X 3 (b) 3 X 2 (c) 4 X 1 (d) 1 X 3

2. (a) 2 6 12

6 16 30

(b)

1 4 5

0 4 7

1 1 9

3. (a)

147

33 (b)

41

11 (c)

3514

18 (d)

3317

87

4. (a)

23

1

25

23

21

(b)

54

62

Mathematics QM 016


SUBTOPIC : 5.1 Matrices

LEARNING

OUTCOMES :

(a) Perform operations on matrices such as multiplication of two

matrices.

(d) Define the transpose of a matrix and explain its properties.

(c) Define symmetric matrix and skew symmetric matrix.

Operation on Matrices

Multiplication of Matrices

The product of two matrices A and B is defined only when the number of columns in A is

equal to the number of rows in B.

If the order of A is m n and the order of B is n p, then AB has order m p.

m n n p m pA B AB

A row and a column must have the same number of entries in order to be multiplied.

1 2 3 nR a a a ... a and

1

2

3

n

b

b

bC

...

b

1 1 2 2 3 3 n nRC a b a b a b ...a b

Example 1

Find

2 11 2 3

3 42 0 5

2 1

Example 2

2 5 4

2 0 5A

and

1 2 3 5

3 2 1 5

5 4 0 7

B

. Find AB.

Mathematics QM 016


Example 3

Let

43

21A and

23

12B

Show that AB ≠ BA.

Properties

(a) A BC AB C ( Associativite)

(b) A B C AB AC ( Distributive)

Transpose Matrix

The transpose of a matrix A, written as AT, is the matrix obtained by interchanging the

rows and columns of A. That is, the i th column of AT is the i th row of A for all i’s.

If m n ijA a then T

n m jiA a

11 12 13

21 22 23

31 32 33 3 3

a a a

A a a a

a a a

then

11 21 31

12 22 32

13 23 33 3 3

T

a a a

A a a a

a a a

Example

Let

3 1

2

1

3

B

then 1 3

2 1 3TB

If

3 3

1 3 3

2 5 4

1 3 5

D

then

3 3

1 2 1

3 5 3

3 4 5

TD

Mathematics QM 016


Example 4

Let

43

21A ,

12

43B and

23

41C .

Show that (a) (A + B )T = A

T + B

T

(b) (BC)T = C

TB

T

Properties of the transpose matrix

(A ± B)T = A

T ± B

T

(AT)

T = A

(AB)T = B

TA

T

(kA)T = kA

T

A symmetric Matrix

A square matrix, A = [ ija ], is symmetric if it is equal to its own transpose,

A = AT and aij = aji .

Example

32

21

2

3

1

cb

ca

ba

A square matrix, A = [ aij ] is a skew symmetrical matrix if A = -AT and aij = -aji where

i j and aii = 0

Example

02

20

031

302

120

Mathematics QM 016


Exercises

1. Let

032

421A ,

43

21B , and

462

154

973

C .Indicate whether the

given product is defined. If so, give the order of the matrix product. Compute the

product, if possible.

(a) AB (b) AC (c) BA

(d) BC (e) CA (f) CB

2. Let

12

04

13

A ,

11

21

12

B and

22

43C .

Find

(a) ATB (b) B

TA (c) (BC)

T (d) (A+B)

T

Answers

1. (a) Not defined

(a) Defined; 2x3 ;19 41 27

18 29 21

(b) Defined; 2x3 ;5 8 4

11 18 12

(c) Not defined

(d) Not defined

(e) Not defined

2. (a)

01

1312 (b)

013

112

(c)

6810

578 (d)

220

355

Mathematics QM 016


SUBTOPIC : 5.2 Determinant of Matrices

LEARNING

OUTCOMES :

(a) Define the minor and cofactor for ija

(e) Discuss the expansion of the cofactor and find the determinant

of a 3 ×3 matrix.

What is the history of determinant of matrices?

Historically, determinants were considered before matrices. Originally, a determinant was

defined as a property of a system of linear equations. The determinant "determines"

whether the system has a unique solution (which occurs precisely if the determinant is

non-zero). In this sense, two-by-two determinants were considered by Cardano at the end

of the 16th century and larger ones by Leibniz about 100 years later.

What can you explain about the determinant and its application.

The determinant is an algebraic operation that transforms a square matrix into a scalar.

This operation has many useful and important properties. For example, the determinant is

zero if and only if the corresponding system of homogeneous equations is singular.

Determinants are used to characterize invertible matrices, and to explicitly describe the

solution to a system of linear equations with Cramer's rule.

Determinant of 2 x 2 Matrices

Given A =

Then determinant A = a b

c d = ad – bc

Example 1

Given A = 2 5

3 8

and B = 3 2

5 2

, find , , ,A B AB BA .

: 4NOTE AB A B BA

a b

c d

Mathematics QM 016


Minor and Cofactor

Let A be n x n matrix,

1. The minor Mij of the element aij is the determinant of the matrix

obtained

by deleting the ith row and jth column of A

2. The cofactor Cij of the element aij is

Cij = (-1)i+j

Mij

Consider the matrix A =

3 4 1

2 4 3

1- 2 1

Minor

M11 is the determinant of the matrix obtained by deleting the first row

and first column from A.

M11 =

341

243

121 =

34

24

= 4

Similarly

M32 =

341

243

121 =

23

11 = 5

Therefore,

If A =

a a a

a a a

a a a

333231

232221

131211

, then

M11 =

3332

2322aa

a a

and M32 =

2321

1311aa

a a

Cofactor

Cij = ( - 1 )i +j

Mij

Then, C11 = (-1)1+1

M11 = 4 and

C32 = (-1)3+2

M32 = -5

Mathematics QM 016


Example 2

A =

3- 3 4 4 0 2- 2- 4- 2 ,

find

i . M12 and C12

ii. M31 and C31

iii. M22 and C22

iv. M23 and C23

_____________________________________________________________________

Determinant of 3 x 3 matrix

Expansion of the cofactor

1 2 1 2

;

, , ..., , , ...,

ij ijA a c

i n and j n

By expanding along the first row

Elements in 1st row : 11 12 13, ,a a a

11 11 12 12 13 13

A a c a c a c

Where , 1 ( )i j

ij ijc m

11 11 12 12 13 13

A a m a m a m

By expanding along first column

Elements in 1st column : 11 21 31, ,a a a

11 11 21 21 31 31 A a c a c a c

11 11 21 21 31 31A a m a m a m

3 1 4

1 2 7 , e ;

5 1 10

) sec

)

Let A find A by xpanding along

a ond row

b first column

Mathematics QM 016


21 21 22 22 23 23( ) ( )

( 1)( 6) ( 2)(10) ( 7)(2)

6 20 14 28

A a m a m a m

b) Expansion along 1st column,( ( 1, 1,2,3)j i

11 11 21 21 31 31A a c a c a c

3( 20 7) ( 1)( 10 4) 5( 7 8)

28

HINT- Choose row or column that has the most zero.

Example 3

Find the determinant of

2 5 1

3 0 1

2 5 4

A

by using expansion of the cofactor

EXERCISES :

1. Find the determinant for these matrices by using the method above:

(a) A=

32

36

(b) B =

652

420

132

2. Find M12 and M33 for matrix B (from Q1).

ANSWERS:

1. (a) | A | = 24, (c) | B | = -44

2. M12 = 8, M33 = 4,

Mathematics QM 016



LEARNING

OUTCOMES : (a) Discuss the properties of determinant

.

Determine |F| if

1 2 3

0 0 0 ,

4 2 1

F then find |F10

|. To find F10

are quite difficult . It is

better to solve the question by using properties of the determinant.

Properties of determinant

1. If any row (or column) of a square matrix A contain only zeroes, then |A|=0

1 2 3

0 0 0 , 0

4 2 1

A A

,

2. If a square matrix B is obtained from a square matrix A by multiplying each

element of any row or column of A by some real number k, then B k A

2 3

2 2 2( 2) 44 5

AB

2B A

thus k = 2

1 0 3

7 0 5 , 0

2 0 1

B B

2 3 2 3, 2

4 5 4 5A A

4 6,

4 5

4 620 24 4

4 5B B

Mathematics QM 016


Example 1

Given the |A| = 5. Hence find |B| , |C| and |D| by using the determinant properties.

1 2 3

2 3 5

3 4 2

A

5 10 15

2 3 5

3 4 5

B

5 10 15

10 15 25

3 4 2

C

5 10 15

10 15 25

15 20 10

D

Note

nkA k A , where A is a square matrix (n x n) and k is a constant.

3. If any two rows (or columns) of a square matrix A are identical, then 0A

3 4 1

4 2 2

3 4 1

A

3 4 1

4 2 2

3 4 1

A = 0

Example 2

Find the determinant of

3 6 9

1 2 3

0 1 1

B

4. If a square matrix B is obtained from a square matrix A by interchanging any two

rows (columns), then B A

2 3

1 4

A

2 3

1 4

A = 8+3 =11

1 4

2 3

B Row 1 and 2 are interchanged

3 8 11 B

B A = -11

Mathematics QM 016


5. If A is a square matrix, then TA A

3 1 4

1 2 7

5 1 10

A 28 A

3 1 5

1 2 1

4 7 10

TA

28TA

28TA A

6. If A and B are square matrices, then AB A B

2 1

3 4

A

1 4

3 2

B

11A 14 B (11)( 14) 154 A B

5 6

9 20

AB

154 AB

AB A B

7. The determinant of an upper (or lower) triangular matrix is the product of its

diagonal entries.

2 0 0

0 3 0

0 0 1

A

3 1 2

0 1 4

0 0 1

B

(2)(3)( 1) A (3)( 1)(1) B

6 1

8. If a square matrix B is obtained from a square matrix A by adding k times the

elements in the ith

row of A to jth

row of A, then B A

1 4

2 1

A 1 4

0 9

B

9 A 9 B

A B

Mathematics QM 016


Exercises

1. If

1 0 1

7 3 2 ,

2 1 2

B x

x

and |B| =4 , find

(a) The value of x (b) | BT | (c) | B

4 |

Answers

1. (a) (b) (c)

Mathematics QM 016



LEARNING

OUTCOMES : (a) Find the adjoint of matrix A

(b) Define the inverse of a matrix

(c) Find the inverse matrix using the adjoint matrix

Adjoint Matrix

Let ijC c be the cofactor matrix of A.

Adjoint of matrix A (adj A) is defined as the transpose of the cofactor matrix that is

adj A = T

T

ij jiC c c

Remember: Cofactor , cij = (-1)i+j mij,

Example 1

Given

1 2 3

3 2 1

1 1 3

A . Find the adjoint of A.

Solution

11

2 42

1 3 c 21

2 33

1 3 c

11

2 42

2 4 c

12

3 45

1 3 c 22

1 30

1 3 c 32

1 35

3 4 c

13

3 21

1 1 c 23

1 21

1 3 c 33

1 24

3 2 c

adj A = CT

11 12 13

21 22 23

31 32 33

m m m

C m m m

m m m

2 5 1

3 0 1

2 5 4

T

adj A

2 3 2

5 0 5

1 1 4

adj A

Mathematics QM 016


Example 2

Given

1 2 2

2 10 5

1 3 3

P . Find the adjoint of P.

Inverse Matrices

There are 3 methods to obtain inverse of matrices:

(a) Adjoint Method -1 1A = adj A

A (Remember! 1 1 A

A)

(b) Use the property AB = k

(c) Elementary Row Operations (ERO)

Finding Inverse by Using Adjoint Method

The inverse of a matrix A is denoted by 1 1A adj A

A

, given that 0A .

If 0A ,

~ A is a non-singular matrix

~ Inverse matrix exists

If 0A ,

~ A is a singular matrix

~ Inverse matrix does not exist

Inverse of a 2 x 2 matrix

Let a b

Ac d

, then 1A is given by

1 1 d bA

c aad bc

Note 1 1A

A

Mathematics QM 016


Example 3

Find the inverse matrix for 3 1

5 4A

Inverse of a 3 x 3 matrix

Example 4

Find the inverse matrix of

1 3 2

0 2 2

2 1 0

B

Properties of Inverse Matrix

1 1

1 1

1 1 1

1

( )

( ) ( )

( )

1

T T

A A

A A

AB B A

AA

AB BA I 1 1andB A A B

1

1

Hence, if

1then

1and also

AB I

A B

B A

Note

If AB I where A and B are square matrices, then B is called the inverse matrix of A and

is written as 1A . Thus 1 1AA A A I

Example 5

Given

1 2 3

2 3 4

1 5 7

A

and

1 1 1

10 4 2

7 3 1

B

. It is known that AB kI , where k is a

constant and I is an 3 3 matrix. Find k and hence deduce 1A .

Mathematics QM 016


Example 6

Given

3 1 1

1 3 1

1 1 3

A

. Find the values m and n such that 2 0A mA nI where I is

3 3 identity matrix and 0 is zero matrix. Use this relation to obtain 1A and show that 3 39 70A A .

_______________________________________________________________________

Exercises

By using adjoint method, find A-1

if it exists

1.

2 1 4

3 2 5

0 1 3

A

2.

2 1 4

3 2 1

5 2 9

A

3.

1 2 0

1 1 0

2 5 1

A

Answers

1. 1

11 1 131

9 6 2225

3 2 1

A

2. 1A does not exist 3. 1

1 2 01

1 1 03

7 1 3

A

Mathematics QM 016


SUBTOPIC : 5.3 Inverse Matrices

LEARNING

OUTCOMES :

(a) apply the elementary row operations to obtain the inverse of

2 x 2 matrix.

There are 3 methods to obtain inverse of matrices:

(a) Adjoint Method -1 1A = adj A

A ( Remember! 1 1 A

A)

(b) Use the property AB = k

(c) Elementary Row Operations (ERO)

Finding Inverse Using Elementary Row Operations.

Given an augmented matrix [ A | I ] with rows Ri, i=1,2,…,m.

The elementary row operations include the following operations:

(a) Interchanging the ith

row and jth

row. i jR R

(b) Multiplying the ith

row with a nonzero constant. *

i jR R (* : is the new row)

(c) Adding a multiple of jth

row to the ith

row. *

i j iR R R

PROCEDURE

STEP 1: Obtain a 1 in the first position on the leading diagonal.

STEP 2: Obtain zeros under 1 in the first column.

STEP 3: Obtain a 1 the second position on the leading diagonal.

STEP 4: Obtain a zero under 1 in the second column.

STEP 5: Obtain a 1 in the third position on the leading diagonal.

STEP 6: Obtain zeros above all the 1’s.

Mathematics QM 016


NOTES: [ A | I ] [ I | A-1

] – As matrix A changes to the Identity matrix, the augmented

Identity matrix changes to the Inverse of matrix A.

Example 1

Find the inverse of the given matrix using elementary row operations.

(a) 1 3

2 5

(b) 2 5

1 3

Solution

Construct an augmented matrix [ A | I ]

Use ERO to find the inverse of A.

[ A | I ] [ I | A-1

]

(a) A = 1 3

2 5

[ A | I ] 1 3 1 0

2 5 0 1

[ A | I ] 1 3 1 0

2 5 0 1

*

2 1 2( 2)R R R

1 3 1 0

0 1 2 1

*

2 2( 1)R R

a b c

d e f

g h i

Step 1

1

Step 2

0

Step 2

0 Step 3

1

Step 4

0

Step 5

1

Step 6

0 Step 6

0

Step 6

0

Mathematics QM 016


1 3 1 0

0 1 2 1

*

1 2 1( 3)R R R

1 0 5 3

0 1 2 1

= [ I | A

-1 ]

A -1

=

12

35

Exercises

1. Find the inverse of the given matrix, if it exists.

(a) A =

73

52

(b) A =

42

03

(c) A =

46

23

2. Find A -1

, if it exists, for the given A using elementary row operations.

When A

-1 exists, verify that A A

-1 = A

-1A = I.

(a) A =

52

63

(b) A =

63

84

(c) A =

01

23

Mathematics QM 016


Answers

1 (a) A -1

=

23

57

(b) A -1

=

4

10

6

1

3

1

(c ) A -1

does not exist

2. (a) A -1

=

13

2

23

5

(b) A -1

does not exist

(c) A -1

=

2

3

2

110

Mathematics QM 016


SUBTOPIC : 5.3 Inverse Matrices

LEARNING

OUTCOMES :

(a) apply the elementary row operations to obtain the inverse of

3 x 3 matrix.

Inverse Matrices

Example 1

Given P =

2 1 1

1 1 1

2 2 1

. Find the inverse of P by using ERO.

Example 2

If B =

2 4 3

1 2 1

2 3 1

. Find B – 1

.

Exercises

1. Find A-1

, if it exists, using elementary row operations

(b) A =

1251

531

321

(c) A =

245

342

013

(d) A =

105

311

217

Mathematics QM 016


2. Given P =

2 4 3

1 2 1

2 3 1

. Find the inverse of P by using ERO.

Answers

1. (a) A-1

=

3

11

3

23

23

3

73

13

3

11

(b) A-1

=

10712

9611

324

(c) A-1

=

17

8

17

5

17

517

19

17

3

17

1417

5

17

1

17

1

2. P-1

=

1 1 0

1 0 1

0 2 1

Mathematics QM 016


SUBTOPIC : 5.4 System Of Linear Equations With Three Variables

LEARNING

OUTCOMES :

(a) Discuss system of linear equations and the types of solutions

namely: unique, inconsistent and infinite solutions.

(b) Write a system of linear equations in the form AX = B

We may have solved linear equation early in the school. Problems involving solving sets

of linear equation are very important in the field of Engineering and Mathematics.

Definition - A system of linear equations is a set of equations with n equations and n

unknowns, is of the form of

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

...

...

...

x n

x n

x x xx n n

a x a x a x b

a x a x a x b

a x a x a x b

The unknowns are denoted by x1,x2,...xn and the coefficients (a's and b's above) are

assumed to be given.

Systems of Linear Equations

Consider the system of linear equations with three unknown x1 , x2 and x3

3333232131

2323222121

1313212111

bxaxaxa

bxaxaxa

bxaxaxa

All the linear systems above can be written as a single matrix equation as below

With A =

333231

232221

131211

aaa

aaa

aaa

, B =

3

2

1

b

b

b

and X =

3

2

1

x

x

x

Thus , the system of linear equation can be written as AX=B

Mathematics QM 016


Example

Puan Lili wants to buy three different flavours of ice cream for her children. For the first

child, she buys ice cream which costs RM2 for a scoop of strawberry, RM3 for a scoop of

vanilla and RM4 for a scoop of chocolate.

For the second child, she buys ice cream which costs RM4, RM3 and RM1 for a scoop of

strawberry, vanilla and chocolate respectively.

Meanwhile, for third child, the costs is RM1, RM2 and RM4 for a scoop of strawberry,

vanilla and chocolate respectively.

She buys x scoops of strawberry ice cream, y scoops of vanilla ice cream and z scoops of

chocolate ice cream for each child.

Find the values of x, y and z if she spends RM34 on the first child, RM28 on the second

child and RM27 on the third child.

Solution

Let x ~ scoop of strawberry’s ice cream

y ~ scoop of vanilla’s ice cream

z ~ scoop of chocolate’s ice cream

Linear equation: 2x + 3y + 4z = 34

4x + 3y + z = 28

x + 2y + 4z = 27

The system of AX = B is called

Consistent (unique and infinitely) if it has solution

Inconsistent if it has no solution

Types of solution of linear system equations

In a 2-dimensional geometry:

(i) Unique

(ii) Infinitely

(iii) Inconsistent (has no solution)

Consistent (has solution)

Mathematics QM 016


(i) Unique Solution

Example 1

Solve the system 5x + 2y = 22

6x – 2y = 0

Solution

5x + 2y = 22 6(2) – 2y = 0

6 2 0

11 22

2

x y

x

x

12

2

6

y

y

There is one intersection point at (2,6).

Thus the system has a unique solution.

Example 2

Solve the system 2x – y = 0

x + y = 1

(ii) Infinite Solution

Example 3

Solve the system y = 3 – 2x

4x + 2y = 6

Example 4

Solve the system x = 2 + y

-2x + 2y = -4

(iii) Inconsistent Solution

Example 5

Solve the system 4x + 6y = 12

6x + 9y = 12

Example 6

Solve the system 2x – y = 0

4x – 2y = 1

Mathematics QM 016


Exercises

1. Determine the type of solution for the following systems

(a) 2x + 3y = 4

x – 3y = 2

(b) 3x – 5y = 1

-6x + 10y = -2

(c) 3x – 5y = 1

-6x + 10y = 2

Answers

1. (a) unique solution

(b) infinitely solution

(c) inconsistent solution

Mathematics QM 016



LEARNING

OUTCOMES : (a) Solve AX=B using Inverse Matrix.

Definition – Assuming we have a square matrix A, which is non-singular ( det(A) does

not equal to zero ), then there exists an nxn matrix A-1

which is called the inverse of A,

such that this property holds:

AA-1

= A-1

A = I where I is the identity matrix.

Using the Inverse Matrix to solve AX = B

If A is a n x n square matrix that has an inverse A-1

, X is a variable matrix and B ia a

known matrix, both with n rows, then the solution of matrix equations AX = B is given by

X = A-1

B

Proof : A X = B ( 3 x 3 square matrix)

A-1

( A X ) = A-1

B

( A-1

A ) X = A-1

B

I X = A-1

B

X = A-1

B

Example 1

Solve the following system of equations by using the inverse matrix

Example 2

Given A =

1 3 9

5 1 3

1 3 7

B =

1 3 0

2 1 3

1 0 1

Find AB and A-1

. Hence, solve the following linear equations.

5 7

3 3 5

9 3 7 1

x y z

x y z

x y z

1 2 3

1 2 3

1 3

3 2 11

3 2 2 10

5

x x x

x x x

x x

Mathematics QM 016


Example 3

Given A =

1 1 2

0 2 2

1 1 3

.

Find A2 – 6A + 11I, with I as an identity matrix 3 x 3. Show that A(A

2 – 6A + 11I) = 6 I,

hence deduce A-1

.

___________________________________________________________________

Exercises

Solve the following system of equations using the inverse matrix

Answers

1x = -2, 2x = 1 and 3x = -3

1 2 3

1 2 3

1 2

9 4 17

2 6 14

6 4

x x x

x x x

x x

Mathematics QM 016



LEARNING

OUTCOMES : (a) Solve AX=B using Gauss-Jordan Elimination method

In the Gauss-Jordan Elimination Method our goal is to use row operations to transform an

augmented matrix into a reduced form. It can be shown that any linear system must have

exactly one solution, no solution, or an infinite number of solutions, regardless of the

number of equations or the number of variables in the system. The terms unique,

consistent, and inconsistent are used to describe these solutions, just as they are for

systems with three variables.

The Gauss-Jordan elimination method is to used to determine these types of solutions

according to the reduced augmented matrix.

The Gauss- Jordan Elimination Method

1. Form the augmented matrix whose first n columns constitute A and whose last

columns form B , symbolically BA .

2. The elementary row operation are used to reduce the augmented matrix to form a

Reduced Augmented Matrix (RAM).

Example 1

Solve the following system of equations using the Gauss- Jordan Elimination method

3 12

0

2 8

x y z

x y z

x y z

[A B] [I X]

Augmented Augmented

MatrixMatrix

Reduced Reduced

Augmented Augmented

MatrixMatrix

COEFFICIENT

MATRIX

COLUMN

MATRIX

IDENTITY

MATRIX

SOLUTION

MATRIX

Mathematics QM 016


Example 2

Ali, Bob and Ravi bought tickets for three separate performances. The table below

shows the number of tickets bought by each of them.

concert orchestral opera

Ali 2 1 1

Bob 1 1 1

Ravi 2 2 1

(a) If the total cost for Ali was RM 122, for Bob RM 87 and for Ravi RM 146,

represent this information in the form of three equations.

(b) Find the cost per ticket for each of the performances using G-J elimination

method.

(c) Determine how much it would cost Hassan to purchase 2 concert, 1 orchestral and

3 opera tickets.

Exercises

Solve the following system of equations by using the Gauss- Jordan Elimination method

1.

7

15632

10 42

321

321

321

xxx

xxx

xxx

2.

22

42

54

31

21

31

xx

xx

xx

Answers

1. 01 x , 32 x and 43 x

2. 2

11 x , 52 x and 33 x

Mathematics QM 016



LEARNING

OUTCOMES : (a) Solve AX=B using Cramer’s Rule.

A system of linear equations can be written in the form of a matrix equation.

Example: The linear equations,

3x 2y z 6

x 4y 3z 5

5x y z 4

Can be written as

3 2 1 x 6

1 4 3 y 5

5 1 1 z 4

The matrix equation AX = B can be solved by,

(a) finding the inverse matrix method,

(b) using the Gauss Elimination method,

(c) using Cramer’s Rule.

Cramer’s Rule

Step 1 : Find the determinant of Matrix A, A

Step 2 : Replacing the column of A with n x 1 matrix B

333231

232221

131211

aaa

aaa

aaa

3

2

1

x

x

x

=

3

2

1

b

b

b

A X = B

Step 3 : Then the solution is given by

x1 =

1 12 13

2 22 23

3 23 33

b a a

b a a

b a a

A

Mathematics QM 016


x2 =

11 1 13

21 2 23

31 3 33

a b a

a b a

b b a

A

x3 =

11 12 1

21 22 2

31 23 3

a a b

a a b

b a b

A

Example 1

Solve the following system using Cramer’s Rule

1 2 3

1 2 3

1 2 3

2 3

3 11

2 3 9

x x x

x x x

x x x

Example 2


1 2 3

1 2 3

1 2 3

2 3

3 8 5 3

2 2 10

x x x

x x x

x x x

Exercises


1. 1 2 33 4x x x

1 2 3

1 2 3

2 15

2 2 5

x x x

x x x

2. 1 2 32x x x 1

1 2 3

1 2 3

x 2x 3x 1

3x 2x 4x 5

Answers

1. 11 x , 42 x and 53 x

2. 11 x , 32 x and 23 x

Nota Math Matrik

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