c1 matrices

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CHAPTER 1: Linear Algebra (MATRICES, VECTORS)

MATRICES, VECTORS

A matrix is defined as an ordered rectangular array of numbers. They can be used torepresent systems of linear equations. Each number in a matrix is called an entry orelement.

Example: Linear system Matrix Form

1!"

##$

$%$&

'#1

'1

'#1

=+

=

=++

xxx

xx

xxx

=

1

#

$

1!"

#$

%$&

'

#

1

x

x

x

(oefficient matrix) augmented matrix

A *

1!"

#$

%$&

=

11!"

##$

$%$&+

A

, -e can sole /ith augmented matrix by calculations /ith discuss later.

Ans/er are '1 =x ) #1

# =x ) 1' =x . 0ometimes /e use notation x)y and to replace

'#1)) xxx .

Matri A!!ition an! S"alar M#lti$li"ation

T/o matrices haing the same ordercan be addedor subtractedas follo/:

2a3

+++

+++=

+

912711141

300211

9714

301

12111

021

=

211813

322

2b3

=

++

++++

=

+

1518

11560

78117

5605241)(1

711

5021

87

6541

1


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2c3

=

=

31

31

1421

0312

12

01

41

32

2d3

=

=

02

02

22

2211

4431

201)(3

21

43

21

21

41

03

Matri m#lti$li"ation by a %"alar

A matrix A can be multiplied by a scalar as belo/:

=

4

1

2

A )

=

=

12

3

6

4

1

2

33A

=

4-

1-

2-

A-

Matri m#lti$li"ation

T/o matrices can be multiplied if the number of columns in the first is the same as the

number of ro/s in the second. 4n other /ords) the product matrix exists.

A 5 * (

nm pn pm

Equal

6rder of the product matrix

Eam$le:

Find 7E /here [ ]321D= and

=

7

6

5

E 8multiplication of ro/s into

columns9

DE = [ ] [ ]3(7)2(6)1(5)7

6

5

321 ++=

[ ]34=

Eer"i%e%:

#


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2a3

53

02

11

110

7412b3

501

547

321

51

42

Sol#tion

2a3

++++

++++=

1(5)1(0)1)0(1(3)1(2)0(1)

7(5)4(0)1)1(7(3)4(2)1(1)

53

02

11

110

741

=

55

3430

2b3

501

547

321

51

427oes not exist 22 33

Tran%$o%ition

=

54

31A Transposition A *

=

53

41AT

Linear systems of equations) auss elimination

'

not equal


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%!#' =+ zyx '## =++ zyx !'# =+ zyx

+

1"

#;

%

1;!

1%#

!#'

!

#;

%

'#1

1%#

!#'

!

'

%

'#1

1##

!#'

1'1# '#' RRRR

( )

1

#;

%

1

1%#

!#'

%'

#;

%

%'

1%#

!#'

%'


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&a#%% Elimination: T'e T'ree Po%%ible Ca%e% o Sy%tem%

The auss elimination can ta=e care of linear systems /ith a unique solution 2see page"3) /ith infinitely many solution 2see example belo/> page $3) and /ithout solutions

2inconsistent system> see example on page ;3

$


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!


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%


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1


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11


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,?an= A * ?an=+

A * n 2number of column3) there is a unique solution.

,?an= A * ?an=+

A @ n 2number of column3) there is a many solution.

1#


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,?an= A @ ?an=+

A ) there is no solution.

1'


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1&


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1"


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1$


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1;


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1!


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1%


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#


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#1


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a%i" Tet: *rey%+ig, EA!-an"e! Engineering Mat'emati"% .t'e!, /iley, 02

c1 matrices

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