Lecture 1 of 11 (Chap 5, Matrices)

8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)

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5.0 MATRICES AND SYSTEM OF

LINEAR EQUATIONS

5.1 MATRICES

5.2 DETERMINANT OF MATRICES

5.3 INVERSE MATRICES

5.4 SYSTEM OF LINEAR EQUATIONSWITH THREE VARIABLES


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LECTURE 1 OF 11

5.0 Matrices And Systems OfLinear Equations

5.1 Matrices


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At the end of the lesson, students

should be able to:

(a) define matrix and equality of matrices.

(b) identify 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

LEARNING OUTCOMES


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The result of EURO 2006

P W D L G PTS

France 3 2 1 0 7 7England 3 2 0 1 8 6

Croatia 3 0 2 1 4 2

Switzerland 3 0 1 2 1 1

Group B

The above standing shows MATRIXform.


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Matrix Definition

A matrix is a rectangular array ofnumbersenclosed between brackets.

The general form of a matrix withm rowsand n columns:


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mnmmm

n

n

n

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211

m rows

n columns


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The order or dimension of a matrix

withm rowsand n columnsis mxn.

ija

The numbers that makes up a matrix

are called itsentries orelements,

and they are specified by their row

andcolumnposition.

Thematrixfor which the entry is in

ith rowandjthcolumnis denoted by

ija


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Let A=

7322

165

(a) What is the order of A?

(b) If A = [ aij] identify a21and a13

Example 1


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(a)Since A has 2 rows and 3 columns,

the order of A is 2 x 3.

(b) The entry a21is in the second

row and the first column.

Thus, a21=

The entry a13is in the first row and the

third column, and so a13=2

1

-2

Solution


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3 3ija

Given A =

Find matrix A if2

ij

ij i ja

j i i j

Example 2


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11a

12a

13a

31a

21a

22a

23a

32

a

33a

1(1) =

1(2) =

2(2) =

1(3) =

2(3) =

3(3) =

1

2(1) + 2 =

2(2) + 3 =

2(1) + 3 =

3

2

4

4

6

9

7

5

975

644321

A

Solution


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Two matrices are equal if they have the same

dimensionand their corresponding entries areequal

12

21Which matrices below are the same?

A = , B = , C =

12

12

21 21

D =

1221

Solution: A = D

Equality of Matrices

Example 3


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Let A =

248

463

b

a

B=

2832

469

d

c

If A = B, find the value of a, b, cand d.

Example 4


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b = -2

c = 0

3 a = 9

a = -6

4b = -8

6 c = 6 2 3d= 8

3d = -6

d = -2

a=-6 , b = -2, c = 0, d= -2

Solution


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Types of Matrices

1. Row Matrixis a (1 x n) matrix [one row]

A = [ a11 a12 a13 a1n]

Example

A = [ 1 2 3 ]

B = [ 1 0 7 8 4 3 5 ]


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11

21

31

1m

aa

a

.

.

.

a

2. Column Matrixis a (m x 1)

matrix [ one column ]

A =

Example

A =

0

4

,B =

7

5

3

2


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3. Square Matrixis a (nxn) matrix which

has the same number of rows ascolumns.

Example

81

31A = , 2 x 2 matrix

B =

132

213

231

, 3 x 3 matrix


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4. Zero Matrixis a (m x n) matrix which

every entry is zero, anddenoted by .

Example

000000

000

O = O =

00

00O =

0000

00

O


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mmmmm

m

m

m

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211

5. Diagonal Matrix

Let A =

The diagonal entries of A are a11,a22 ,.,amm


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A =

30

02,B =

300

020

001

,C =

b

a

00

000

00

Example

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


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6. Identity Matrix is a diagonal matrix

where all its diagonal entries are 1 anddenoted byI.

10

01

100

010

001

= I2x2= I3x3

Example


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7. Lower Triangular Matrixis a square matrix

and aij= 0 for i < j

323

023

001

A = ,B =

edc

fb

a

0

00

333231

232221

131211

aaa

aaa

aaa

Example

8 Upper Triangular Matrix is a square matrix


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8.Upper Triangular Matrixis a square matrix

and aij= 0 for i > j

300

420

321

P = R =

f

ed

cba

00

0

333231

232221

131211

aaa

aaa

aaa

Example


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Addition And Subtraction Of Matrices

For mx nmatrices

A = and B =

A + B = C = where

A B = D = where

]a[ij

ijijij bac

.ijijij bad

]b[ij

]c[ij

]d[ij

Operations on Matrices


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NOTE

The additionor subtraction

of two matrices with differentordersis not defined.

We say the two matrices areincompatible.


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43

21A

,

65

34B

2

1C

.

FIND :

(a) A + B (b) A

B

(c) A + C

Example 5


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(a) A + B =

43

21 +

65

34

64)5(3

3241

=5 5

2 10

Solution


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(b) A

B =

43

21

-

65

34

=

64)5(33241

=

28

13


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Since matrix A is of order

2 x 2 and matrix C is of order

2 x 1, the matrices have differentorders, thus A and C are

incompatible.

4321

+ 21c) A + C =


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ijcaijb

then

][ ijaA

where]b[cA ijIf cis a scalarand

Scalar Multiplication


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2 4

8 5

6 7

A

Given

1

2AFind

Example 6

Solution


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1 12 2

1 12 2

1 1

2 2

2 4

1 8 52

6 7

A

2

7

25

3

4

21

=

Solution


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Let and

calculate 3A 2B

3541A

2463B

Example 6

Solution


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35 413 24 632

48

126

915

123

57

03

=

=

=

Solution

3 2A B


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Types of Matrices

1. Row Matrix

2. Column Matrix

3. Square Matrix

4.Zero Matrix

5. Diagonal Matrix

6. Identity Matrix

7. Lower/Upper Triangular Matrix


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NOTE

The additionor subtraction

of two matrices with differentordersis not defined.

We say the two matrices are

incompatible.


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1. Identify the order of the given matrix

654

321(a) ( b )

10

01

01

(c)

d

c

b

a

(d) kji

Exercises:


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2

2

ij

( i j ) ,i j b = ij ,i j

( i j ) ,i j

2(a) Find matrix A = [aij]2x3

if aij= i2j + j2i

(b) Find matrix B = [bij]3x3


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53

21A

94

12B

456

321

000

000

42

004

33

62XX

3. Simplify the given quantity for

and

(a) A + B (b) AB

(c) 2A5B (d) 3A + 2B

=

(b) -2

4. Solve the given equation for

the unknown matrix X.

(a) 2X+


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2 6 12

6 16 30

1 4 5

0 4 7

1 1 9

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

(b)

2. (a)

ANSWERS


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147

33

41

11 3. (a) (b)

3514

18

3317

87(c) (d)

23

1

25

23

21

62

4. (a)

(b)

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Lecture 1 of 11 (Chap 5, Matrices)