of 25
7/23/2019 9 - Solving Linear Systems
1/25
2014 Baylor UniversitySlide 1
Fundamentals of Engineering AnalysisEGR 1302 The Inverse Matrix
7/23/2019 9 - Solving Linear Systems
2/25
2014 Baylor UniversitySlide 2
Solving Systems of Linear Equations
The Inverse atri!
"n!nand Bn!nare Square atri#es of the same $rder%
" & B ' In
B & " ' In
B is #alled (The Inverse of ")
B ' "*1
" & "*1
' I
in "lge+ra, the equivalent is 11
=
x
x
7/23/2019 9 - Solving Linear Systems
3/25
2014 Baylor UniversitySlide -
Solving Systems of Linear Equations
Sum of .rodu#ts
The same equation #an re/resent any$rder%
=
2
1
2
1
2221
1211
d
d
x
x
aa
aa
1212111 dxaxa =+
2222121 dxaxa =+
dxA =
7/23/2019 9 - Solving Linear Systems
4/25
2014 Baylor UniversitySlide 4
Solving this System for !1, !2
Be#omes
Su+tra#t
22a
12a
1212111 dxaxa =+
2222121 dxaxa =+
1222221212211 daxaaxaa =+
2122122211221 daxaaxaa =+
212122112212211 !" dadaxaaaa =
211121212212211 !" dadaxaaaa +=
!" 12212211
2121221
aaaa
dadax
=
!" 12212211
2111212
aaaa
dadax
+=
7/23/2019 9 - Solving Linear Systems
5/25
2014 Baylor UniversitySlide
Solving this System for !1, !2 #ont%
Sum of .rodu#ts
" ne3 atri! ()
fa#tor out
the denominator
+
=
= 211121
212122
122122112
1
!"
1
dada
dada
aaaax
x
x
=
2
1
1121
1222
122122112
1
!"
1
d
d
aa
aa
aaaax
x
7/23/2019 9 - Solving Linear Systems
6/25
2014 Baylor UniversitySlide 5
6e no3 have a solution for the Inverse of a 2!2 atri!
The Solution to
dxA =
=
2
1
1121
1222
122122112
1 !"
1dd
aaaa
aaaaxx
dAxAA 11 =
xIxAAwhere 1
=
dAx 1=
=
1121
1222
12212211
1
!"
1
aa
aa
aaaaA
7/23/2019 9 - Solving Linear Systems
7/25 2014 Baylor UniversitySlide 7
The 8eterminant
The 8eterminant of " '
=
1121
1222
12212211
1
!"
1
aa
aa
aaaa
A
=
2221
1211
aa
aaA12212211 aaaa
12212211det aaaaA =
7/23/2019 9 - Solving Linear Systems
8/25 2014 Baylor UniversitySlide 9
:ules for ;inding the Inverse of a 2!2 atri!
Rule 1: Swap the Main Diagonal
Rule 2: Change Signs on
the Back Diagonal
Rule 3: Divide by the Deterinant
=
1121
1222
12212211
1
!"
1
aa
aa
aaaaA
=
1121
1222
12212211
1
!"
1
aa
aa
aaaa
A
=
2221
1211
aa
aaA
7/23/2019 9 - Solving Linear Systems
9/25 2014 Baylor UniversitySlide umeri#al E!am/le
Be#omes
=
213$
!!13"$2"11A
$32 21 =+ xx
2$21
=+ xx
=
2
$
$1
32
2
1
x
x
=
2
$
11%211%1
11%311%$1A
=
11%&
11%10x
7/23/2019 9 - Solving Linear Systems
11/25 2014 Baylor UniversitySlide 11
This is the
definition of the
inverse for any
matri!=
The ?eneral larger than 2!2 Solution to
The 2!2 #ase@
"lready sho3n that this is det" all this the (adAoint) or adA"
The general #ase@
8efinition of adAoint@
dxA =
dAx 1=
=
1121
1222
12212211
1
!"
1
aa
aa
aaaaA
A
CfA
A
AA
T
==!det"
!"ad'1
=
1112
2122
aa
aaCfA
==
1121
1222!"ad'
aa
aaCfAA T
7/23/2019 9 - Solving Linear Systems
12/25 2014 Baylor UniversitySlide 12
Gaussian Elimination ( )olutions of *inear
)ystems
Fundamentals of Engineering Analysis
EGR 1302
7/23/2019 9 - Solving Linear Systems
13/25 2014 Baylor UniversitySlide 1-
The >eed for a ?eneral Solution to Linear Systems
Unique Solution, all
#ross at the same
/oint, the (solution)
Det ! "
#e need a ethod o$
$inding a general solution
when the coe$$icient
atri% & is Singular'
the system
of equations
1x
2x
3x
+2 321 =+ xxx
232 321 =+ xxx ,2 321 =+ xxx
7/23/2019 9 - Solving Linear Systems
14/25 2014 Baylor UniversitySlide 14
?aussian Elimination * " general solution ethodology
Elementary :o3 $/erations@
1% ulti/ly +y a #onstant *
2% S3a/ t3o ro3s *
-% :e/la#e a ro3 +y adding to it another &ro3 *
6e 3ill use three eleentary row operationsto solve this set of
linear equations +y ?aussian Elimination%
Same as multi/lying +oth
sides of an equation
$rder doesnCt matter
Same as elimination
ii rkr
-=
ijji rrrr .
jii rkrr - +=
+2$
+3212
321
321
321
=+
=+=++
xxx
xxxxxx
7/23/2019 9 - Solving Linear Systems
15/25 2014 Baylor UniversitySlide 1
Using Elementary :o3 $/erations
to Solve +y ?aussian Elimination
Ste/ 1@ Use :ule - to eliminate from ro3s 2 D -@
1% ee/ :o3 1 the same
-%
2%
Ste/ 2@ Use :ule - to eliminate from ro3 -@
1% ee/ :o3 1 the same
-%
2% ee/ :o3 2 the same
Ste/ -@ Use :ule 1 to redu#e all #oeffi#ients to 1@
1% ee/ :o3 1 the same
-%
2%
+2$
+32
12
321
321
321
=+
=+
=++
xxx
xxx
xxx
&320
,+0
12
32
32
321
=+
=++
=++
xx
xx
xxx
1x
13
-
3 rrr =12
-
2 2 rrr +=
+
$/
+
2300
,+0
12
3
32
321
=+
=++
=++
x
xx
xxx2x
23
-
3+
2rrr =&320
,+0
12
32
32
321
=+
=++
=++
xx
xx
xxx
200
+
,
+
10
12
3
32
321
=++
=++
=++
x
xx
xxx
3-
323+ rr =+
$/+
2300
,+0
12
3
32
321
=+
=++
=++
x
xx
xxx
2
-
2+
1rr =
12 =x
11=x
7/23/2019 9 - Solving Linear Systems
16/25 2014 Baylor UniversitySlide 15
The "ugmented atri!
#an +e re/resented as
(augmented) matri!
Ste/ 1 * 1% ee/ :o3 1 the same
-%
2%
Ste/ 2 * 1% ee/ :o3 1 the same
-%
2% ee/ :o3 2 the same
Ste/ - * 1% ee/ :o3 1 the same
-%
2%(Row )chelon *or+
+2$1+132
1121
+2$+32
12
321
321
321
=+=+
=++
xxxxxx
xxx
13-
3 rrr =
12
-
2 2 rrr +=
&320
,1+0
1121
23
-
3
+
2rrr =
+
$/
+
2300
,1+0
1121
3
-
3
23
+rr =
2
-
2
+
1rr =
2100+
,
+
110
1121
7/23/2019 9 - Solving Linear Systems
17/25 2014 Baylor UniversitySlide 17
:edu#ed :o3 E#helon ;orm of the "ugmented atri!
Using (Ba#3ards Su+stitution)on the :o3 E#helon ;orm
,dentity Matri%
(Reduced Row )chelon *or+
$+serve dire#tly that
31
-
1 rrr =
32-
2 +1 rrr =
2100+
,
+
1
10
1121
2100
1010
1021
21
-
1 2rrr =
2100
1010
1001
2
1
1
3
2
1
=
=
=
x
x
x
7/23/2019 9 - Solving Linear Systems
18/25 2014 Baylor UniversitySlide 19
Using the TI*9< to do ?aussian Elimination
>ote that #al#ulator #om/utes a different :E; result,+y using a different algorithm, +ut the ans3er is still #orre#t%
Save the augmented matri! as varia+le (!-4)
Reduced Row )chelon *or -
use the $unction (rre$./
Row )chelon *or -
use the $unction (re$./
TI*9< :esult anual :esult
+2$1+132
1121
2100+
,
+
110
1121
7/23/2019 9 - Solving Linear Systems
19/25 2014 Baylor UniversitySlide 1ote@ det"'0
0his syste has no solution
(inconsistent+
22
,33
3
321
321
321
=+
=+
=+
xxx
xxx
xxx
2211,313
3111
1320
$/$0
3111
13-
3
12
-
2 3
rrr
rrr
=
=
23
-
3
2
1rrr =
1000
$/$0
3111
10 3=x
7/23/2019 9 - Solving Linear Systems
20/25 2014 Baylor UniversitySlide 20
The Infinite Solution
hange the System
Same :o3
o/erations
0his syste has in$inite
solutions depending
on the value o$
iplies
det" still equals Fero
2211
,313
1111
=3x
1320
2/$0
1111
13
-
3
12
-
2 3
rrr
rrr
=
=
0000
2/$0
1111
23
-
32
1rrr =
+
=
2
3
2
12
1
2
3
3
2
1
x
x
x
00 3=x
2
3
2
1.2/$ 22 +==+ xx
2
1
2
3.1!
2
3
2
1" 11 ==++ xx
7/23/2019 9 - Solving Linear Systems
21/25 2014 Baylor UniversitySlide 21
The Three ?eneral Solutions
1' niue ! X3e%ists as a single value
2' 4one ! 4o X3e%ists
3' ,n$inite ! X3e%ists as any value
k100
10
1
0det A
k000
10
1
0det =A
000010
1
0det =A
7/23/2019 9 - Solving Linear Systems
22/25 2014 Baylor UniversitySlide 22
?ra/hi# E!am/les of the Three ?eneral Solutions
rre$./
rre$./
rre$./
in$inite
uniue
no solution
planes
parallel
never
intersect
all planes
intersect
on the sae line
single point
o$ intersection
12+3$
,/312102102
=+
=+=++
zyx
zyxzyx
22
,33
1
=+
=+
=+
zyx
zyx
zyx
22
2233
1
=+
=+
=+
zyx
zyx
zyx
,&,&06
,&,103
,&,
,+1
100
010001
0000
10
01
21
23
23
21
1000
010
001
23
21
7/23/2019 9 - Solving Linear Systems
23/25 2014 Baylor UniversitySlide 2-
E!am/le of the Three ?eneral Solutions
?iven 5
1% Unique Solution
-% Infinite Solutions
2% >o Solutions
Unno3ns@
1a
1
1
+
=
a
bz
ba11
3212
2321
13
-
3
12
-
22
rrr
rrr
=
=23
-3 rrr =
+
1100
1$30
2321
ba
!1"3det += aA
1=b1=a0
0=z
1b1=a0
kz=
z
y
x
7/23/2019 9 - Solving Linear Systems
24/25
2014 Baylor UniversitySlide 24
The Gomogeneous Set of Linear Equations
S/e#ial ase
$nly 2 /ossi+le solutions@
, im/lying infinite solutions
6hen k', infinite solutions e!ist,
other3ise, there is no solution%
rre$./
, a trivial solution, orEither
03
02
0
321
21
321
=+
=+
=++
xkxx
xx
xxx
031
0021
0111
k
0,00
0110
0111
k
,det =kA
0=xA
=3x
0det =A
0=x
== 22 .0 xx
2.0 11 ==++ xx
7/23/2019 9 - Solving Linear Systems
25/25
S
Huestions