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is calculatedby D:atla22-an(7zt For the third-ordercase,the determinant can be computedas
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where the 2 by 2 determinantsare called minors Determinonts
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Problem Stotement Computevaluesfor the determinants the systems of represented in Figs9l and92 S o l u t i o n F o r F i g 9 1:
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2--l 1
: - ; ( ll ) - t l ; l :/01 \
L \L,/
ForFig92b: D:l l-1 2
r - : - t I : 2 )( 1(-l):0
GAUSSELIMINATION
For Fig 92c: D_
)' ,', _:l :-
___1)_ll_l:_004
In the foregoing example,the singular systemshad zero determinants Additionally, (Fig 92c) has a detenninant the resultssuggest that the systemthart ahnostsingular' is that is close to zero These ideas will be pursuedfurther in our subsequent discussion illof conditioning ChapI l in that eachunknownin a system linearalgebraic equaCromer's Rule This nrle states of tions may be expressed a fraction of two determinants as with denorninator andwiththe D numeratorobtained from D by replacing the colurnn of coefficientsof the unknown in q u e s t i o n y t h e c o n s t a n tb r b 2 , b , , F o r e x a m p l ef,o r t h r e ee q u a t i o n s ,1 w o u l d b e b s x computed as lbr oe 4r:i I bz azz azt I I tr^ a1 4 l ,
,,:
EXAMPLE Cromer's 92 Rule
rule to solve ProblemStotement Use Cramer's 0 3 ; r r 0 5 2 x 21 + 05rr + r:: -001
067 0 1 r r + 0 3 x 2 * 0 5 1 3: - 0 4 4
x2f l9rj:
canbe evaluated [Eq (91)]: as Solution The determinantD
a:o:lol: l l-o"l3i A:31'l3i -ooo22 o:l:
The solutioncan be calculated as
r I -
-00r 052 I 067 r 19 -044 03 05
-40022 103 -001 | 0I I I
003278 - 1 4 9 :
i or 067 re I
-044 05| -00022 -001 103 052 ,00022
00649 _ _)a < -00022
I os r 06l I or 03 *044
-0043s6
: 198
SOLVING SMALT NUMBERS EAUATIONS OF
can The ae-' Function The determinant be computeddirectly in MATLAB with the det function For example,using the systemfrom the previousexample: >> A-103 052 I;05
>> D= 0 0422 D=det (A)
ii ti
1 19;01
03 051;
Cramer'srule can be appliedto compute{i as in >> A(:,1)= t-00I;061 ; -0441 -00100 06700 -04400
>> x1=det
X1 =
05200 10000 03000
10000 19000 05000
(A) /D
-149000 For more than three equations,Cramer's rule becomesimpractical because,as the number of equationsincreases, determinants time consumingto evaluateby hand the are (or by computer)Consequently, more efficient alternatives usedSomeof thesealterare nativesarebasedon the last noncomputer solutiontechnique coveredin Section913-the elimination of unknowns
913 Eliminotion Unknowns of
The elimination of unknownsby combiningequations an algebraicapproach is that can be illustrated a setof two equations: for
a 1 1 x 1* a 1 2 X 2 : 1 1 1 aZtxtla22x2:fi2
(e2) (e3)
The basic strategyis to multiply the equations constants that one of the unknowns by so will be eliminatedwhen the two equationsare combinedThe result is a single equation that can be solvedfor the remainingunknownThis valuecan thenbe substituted into either of the original equations computethe other variable to For example, Eq (92)might be multipliedby a21andEq (93)by all to give
aztTltxt I a21A12X: AZlbt 2 attbz
(e4)
(95)
: a11a21Xy a11a22x2 {
SubtractingEq (94) from Eq (95) will, therefore,eliminatethe x1 term from the equations to yield : a t t a 2 2 x- a y l a l 2 x z a t t b Z- a y b t 2 which can be solvedfor
a 1 1 b 2a21b1
ELIMINATION GAUSS
(96)can thenbe substituted Eq (92),which can be solvedfbr Equation into
a 2 2 b 1- u p b 2
d 1 1 4 2 202yAp
Noticethat Eqs(96)and (97)follow directlyh'om Cramer's rule:
th' anl
,_t-_-t_
attl
aybl - a12lt2
0|tt)2 A2jal
Il, , , a n l l
1O t t A))
l o " ut' l b
1,3 _1____J___i_I
lrr'
d 1 1 b 21
a21b1
lQtr tl
I clrr
an J
tltt I
to with rnorethan two or three The eliminationof unknownscan be extended systems make eqLrations However, the numerouscalculationsthat are requiredfor larger systems in 9,2, the methodextremelytediousto irnplementby handHowever,as described Section lbr the techniquecan be formalizedand readily programrned the conlputer
NAIVE GAUSSELIMINATION
In Section913,the eliminationof unknownswas usedto solve a pair of simultaneous (Fig93): of equations procedure The consisted two steps 1 2 were manipulated elinrinate of the unknownstiom the equations, to one The equations of this elimination stepwas that we had one equationwith one unknown The result into could be solveddirectly andthe l'esultback-substituted Consequently, equation this solvefor the remainingunknown one of the original equations to
a This basicapproach can be extended largesetsofequationsby developing systemto and Gausselimination atic schemeor algorithrnto eliminateunkrrowns to back-substitute is the most basicof theseschemes tor This sectionincludesthe systematic techniques forward eliminationandbacksubfor are stitutionthatcompliseGauss elimination Althoughthese techniques ideallysuited implementationon computers,some modifications will be required to obtain a reliable algorithrnIn particular,the computerprogram must avoid division by zero The tbllowing method is called "naive" Gausselimination beciiuseit does not avoid this problem Section 93 will deal with the additional featuresrequired for an et'fectivecomputer progranl to The approachis designed solve a generalsetof n equations:
d 1 1 1 1 q s z x z l 4 r - t x :* ' " * ayll * ( r 2 X 2I d z r j r : * ' I ay,,x,,: $t (984)
' I a 2 , r x ,: S ,
(98r)
1a,-r- :
(98c)
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