matrix in computing

7/26/2019 Matrix in Computing

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Matrix for Business


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Matrices

A matrix consisting of m horizontal rows and n

vertical columns is called anm

n matrix

or amatrix of size mn.

For the entry aij, we call i the row subscript andjthe column subscript.

mnmm

n

n

aaa

aaa

aaa

...

......

......

......

...

...

21

21221

11211


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a.he matrix has size .

b.he matrix has size .

c.he matrix has size .

d.he matrix has size .

Chapter 3: Matrix Algebra

Matrices

Example 1 Size of a Matrix

[ ]!21 "1

#$

1%

&1

2"

[ ]' 11

1112&

(&%11$

#2'"1

%"


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Eqalit! of Matrices

)atrices A * +aij and " * +bij are eqalifthey have the same size and aij * bijfor

each iandj.#ra$spose of a Matrix

A tra$spose matrix is denoted by A.

If , find .

-olution

/bserve that

TA

=

&%#

"21A

=

&"

%2

#1TA

( ) AA TT =


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Page %

Matrix A&&itio$ a$& Scalar Mltiplicatio$

Example 1 Matrix A&&itio$

Matrix A&&itio$ Sm A 0 " is the m nmatrix obtained by adding

corresponding entries of A and ".

a.

b. is impossible as matrices are not of the same

size.

+

1

2

#"

21

=

++

++

=

+

68

83

08

0635

4463

2271

03

46

27

65

43

21


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Example 2 Matrix Sbtractio$

a.

=

+ +=

1"

!(

(#

"2!"

11##

2&&2

"!

1#

2&

2"

1#

&2


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Page (

Example 3 )ema$& *ectors for a$ Eco$om!

Demand for the consumers is

For the industries is

What is the total demand for consumers and the

industries?

-olution

otal

[ ] [ ] [ ]12'!%2" "21 === DDD

[ ] [ ] [ ]!%"!(!2!#1! === SEC

DDD

[ ] [ ] [ ] [ ]1(2%'12'!%2""21

=++=++ DDD

[ ] [ ] [ ] [ ]12&%!!%"!(!2!#1! =++=++ SEC

DDD

[ ] [ ] [ ]"!"1%'12&%!1(2%' =+


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Page +

Scalar Mltiplicatio$

roperties of -calar )ultiplication

Sbtractio$ of Matrices

roperty of subtraction is ( )AA 1=


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Page ,

To multiply matrices Aand B

look at their dimensions

)- 34 -A)4

-564 /F 7/89

5f the number of columns ofAdoes note:ual the number of rows of Bthen theproductABis undefined.

pnnm


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Multiplication of Matrices#he mltiplicatio$ of matrices is easier sho.$ tha$ pt

i$to .or&s/ 0o mltipl! the ro.s of the first matrix

.ith the colm$s of the seco$& a&&i$g pro&cts

Find AB

irst .e mltipl! across the first ro. a$& &o.$ the

first colm$ a&&i$g pro&cts/ e pt the a$s.er i$

the first ro. first colm$ of the a$s.er/

=

140123A

=

13

31

42

B

( )23( ) ( ) ( )1223 +( ) ( ) ( ) ( ) ( ) ( ) 5311223 =++


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Find AB

e mltiplie& across first ro. a$& &o.$ first colm$

so .e pt the a$s.er i$ the first ro. first colm$/

o. .e mltipl! across the first ro. a$& &o.$ the seco$&

colm$ a$& .e5ll pt the a$s.er i$ the first ro. seco$&colm$/

o. .e mltipl! across the seco$&ro. a$& &o.$ the first

colm$ a$& .e5ll pt the a$s.er i$ the seco$& ro. firstcolm$/

o. .e mltipl! across the seco$&ro. a$& &o.$ the

seco$& colm$ a$& .e5ll pt the a$s.er i$ the seco$& ro.seco$& colm$/

Notice the sizes of Aand Band the size of the product AB.

= 140123

A

=

13

31

42

B

= 5AB ( ) ( )43( ) ( ) ( ) ( )3243 +( ) ( ) ( ) ( ) ( ) ( ) 7113243 =++

= 75AB ( ) ( )20( ) ( ) ( ) ( )1420 +( ) ( ) ( ) ( ) ( ) ( ) 1311420 =++

=1

75AB ( ) ( )40( ) ( ) ( ) ( )3440 +( ) ( ) ( ) ( ) ( ) ( ) 11113440 =++

=111

75AB


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o. let5s loo6 at the pro&ct "A/

2332 ca$mltipl!

sizeofa$s.er

across first row as

we go down first

colun!

across first row as

we go down

second colun!

across first row as

we go down third

colun!

across second row

as we go down

first colun!

across second row

as we go down

second colun!

across second row

as we go down

third colun!

across third row

as we go down

first colun!

across third row

as we go down

second colun!

across third row

as we go down

third colun!

Completel! &iffere$t tha$AB7

Commuter's Beware!

=

6

BA

=

126

BA

=

2126

BA

= 3

2126

BA

= 143

2126

BA

= 4143

2126

BA

=

"

4143

2126

BA

=

10"

4143

2126

BA

=

410"

4143

2126

BA

=

13

31

42

B = 140123

A

BAAB

( ) ( ) ( ) ( ) 60432 =+( ) ( ) ( ) ( ) 124422 =+( ) ( ) ( ) ( ) 21412 =+( ) ( ) ( ) ( ) 30331 =+( ) ( ) ( ) ( ) 144321 =+( ) ( ) ( ) ( ) 41311 =+( ) ( ) ( ) ( ) "0133 =+( ) ( ) ( ) ( ) 104123 =+( ) ( ) ( ) ( ) 41113 =+


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Matrix Mltiplicatio$

Example 3 Matrix Pro&cts

a.

b.

c.

d.

[ ] [ ]326

5

4

321 =

[ ]

=

183122

61

6132

1

=

1047

014

1135

212

312

201

401

122

031

++++

=

2222122121221121

2212121121121111

2221

1211

2221

1211

babababa

babababa

bb

bb

aa

aa


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Example % Cost *ector

i!en the "rice and the #uantities, calculate the total

cost.

-olutionhe cost vector is

[ ]#"2=$

9ofunits

3ofunits

Aofunits

11

%

'

=%

[ ] [ ]'"11

%

'

#"2 =

=$%


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Exercise Cost *ector

i!en the "rices &in dollars "er unit' for three

te(tboo)s are re"resented b* the "rice !ector. A

uni!ersit* boo)store orders these boo)s in the

#uantities +i!en b* the column !ector %. find the total

cost &in dollars' of the "urchase.

22,(#".'%

[ ]50.2875.3425.26=P

=

175

325

250

Q

Ch t 3 M t i Al b


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Example ( Associati8e Propert!

If

com"ute A"C in to a*s.

-olution 1 -olution 2

;ote that A


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Example 11 Matrix 9peratio$s $8ol8i$g a$& 9

If

com"ute each of the folloin+.

-olution

!!

!!

1!

!1

#1

2"

1!

"

1!

1

%

1

%

2

=

=

=

= -IBA

"1

22

#1

2"

1!

!1

a.

=

= AI

( )

=

=

=

&"

&"

2!

!2

#1

2""

1!

!12

#1

2""2"b. IA

-A- =

=

!!

!!

#1

2"c.

IAB =

=

=

1!

!1

#1

2"d.

1!

"

1!

1

%

1

%

2



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Example 13 Matrix orm of a S!stem ;si$g Matrix Mltiplicatio$

Write the s*stem

in matri( form b* usin+ matri( multi"lication.

-olution5f

then the single matrix e:uation is

=+

=+

'"(

#%2

21

21

((

((

=

=

=

'

#

"(

%2

2

1B

(

(.A

=

=

'

#

"(

%2

2

1

(

(

BA.


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Example , style houses, seven 9ape 9od stylehouses, and nine colonial style houses.his order represented by

he raw materials that go into each type ofhouses are steel, wood, glass, paint, andlabor shown in the matrix 7.


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-uppose that the contractor want to ?nowhow many amount of each raw material

needed to fulfill the orders. he we shouldcompute the matrix @7.

he contractor also interested in the costsfor the materials for each, shown in matrix9

[ ]34"143236451128=QR


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hen, the cost of each type of house is given bymatrix

9onse:uently, the total costs of materials forranch style '%,(%!B cape 9od style (1,%%!Band colonial style '1,&%!

he total cost of raw materials for all the houses

is

he total costs is 1,%$#,$%!

=

=

'1&%!

(1%%!

'%(%!

1%!!

1%!

(!!

12!!

2%!!

1"%(2%&

21$121('

1''1&2!%

/C

( ) [ ] [ ]"50#5"4#1

71650

81550

75850

"75 =

== RCQQRC

2 2


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$8erses 2x2 Matric

Example 1 $8erse of a Matrix

Chen matrix A" * , Ais an inverse of "

and A is invertible/

=

2153B

= 31

52A



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$8erses i$& matrix :

ad>bc * 5A5*determinant A



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$8erses i$& matrix :

= 31

52

A


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4xercises

Find its inverse

3 3


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$8erses 3x3

Find the minor of )atric A

D 4liminate the first row andfirst coloumn, get 5)115

5)115 * a22xa""E a2"xa"2


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;ow do the adoint, that is the transposeof matrix cofactor )

hen, do the determinant


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4xample inverse matric "x"

A *

G *

400

030001

300

040

0012


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300

040

0012


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A>1

4$100

03$10

001


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Find its inverse

103

010

207

@ i no 1


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@uiz no.1 Hababe?a property accepted orders for five ranch>style

houses, two 9ape 9od style houses, and four colonial

style houses. his order represented by

he raw materials that go into each type of houses aresteel, wood, glass, paint, and labor shown in the matrix

7.

he contractor also interested in the costs for the

materials for each, shown in matrix 9

sing matrix multiplication, compute the total cost of raw

materials

matrix in computing

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