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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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Page '
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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Page 1-
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$/
Notice the sizes of Aand Band the size of the product AB.
= 140123
A
=
13
31
42
B
= 5AB ( ) ( )43( ) ( ) ( ) ( )3243 +( ) ( ) ( ) ( ) ( ) ( ) 7113243 =++
= 75AB ( ) ( )20( ) ( ) ( ) ( )1420 +( ) ( ) ( ) ( ) ( ) ( ) 1311420 =++
=1
75AB ( ) ( )40( ) ( ) ( ) ( )3440 +( ) ( ) ( ) ( ) ( ) ( ) 11113440 =++
=111
75AB
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Page 12
o. let5s loo6 at the pro&ct "A/
2332 ca$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$AB7
Commuter's Beware!
=
6
BA
=
126
BA
=
2126
BA
= 3
2126
BA
= 143
2126
BA
= 4143
2126
BA
=
"
4143
2126
BA
=
10"
4143
2126
BA
=
410"
4143
2126
BA
=
13
31
42
B = 140123
A
BAAB
( ) ( ) ( ) ( ) 60432 =+( ) ( ) ( ) ( ) 124422 =+( ) ( ) ( ) ( ) 21412 =+( ) ( ) ( ) ( ) 30331 =+( ) ( ) ( ) ( ) 144321 =+( ) ( ) ( ) ( ) 41311 =+( ) ( ) ( ) ( ) "0133 =+( ) ( ) ( ) ( ) 104123 =+( ) ( ) ( ) ( ) 41113 =+
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Chapter 3: Matrix Algebra
Matrix Mltiplicatio$
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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Page 14
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 =
=$%
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Page 1%
Exercise Cost *ector
i!en the "rices &in dollars "er unit' for three
te(tboo)s are re"resented b* the "rice !ector. A
uni!ersit* boo)store orders these boo)s in the
#uantities +i!en b* the column !ector %. find the total
cost &in dollars' of the "urchase.
22,(#".'%
[ ]50.2875.3425.26=P
=
175
325
250
Q
Ch t 3 M t i Al b
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Page 1'
Chapter 3: Matrix Algebra
Matrix Mltiplicatio$
Example ( Associati8e Propert!
If
com"ute A"C in to a*s.
-olution 1 -olution 2
;ote that A
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Chapter 3: Matrix Algebra
Matrix Mltiplicatio$
Example 11 Matrix 9peratio$s $8ol8i$g a$& 9
If
com"ute each of the folloin+.
-olution
!!
!!
1!
!1
#1
2"
1!
"
1!
1
%
1
%
2
=
=
=
= -IBA
"1
22
#1
2"
1!
!1
a.
=
= AI
( )
=
=
=
&"
&"
2!
!2
#1
2""
1!
!12
#1
2""2"b. IA
-A- =
=
!!
!!
#1
2"c.
IAB =
=
=
1!
!1
#1
2"d.
1!
"
1!
1
%
1
%
2
Chapter 3: Matrix Algebra
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Page 1+
Chapter 3: Matrix Algebra
Matrix Mltiplicatio$
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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Page 2-
-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
Chapter 3: Matrix Algebra
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Chapter 3: Matrix Algebra
$8erses i$& matrix :
ad>bc * 5A5*determinant A
Chapter 3: Matrix Algebra
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Chapter 3: Matrix Algebra
$8erses i$& 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
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