Revision 3 by 3 Matrices and Equations

8/13/2019 Revision 3 by 3 Matrices and Equations

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Further Pure 1

Revision Topic 7: 3 by 3 matrices and systems of equations

The OCR syllabus says that candidates should:

a) recall the meaning of the terms singular and non-singular as applied to square matrices, and,

for ! matrices, e"aluate determinants and find in"erses of non-singular matrices#b) formulate a problem in"ol"ing the solution of equations in un$no%ns as a problem

in"ol"ing the solution of a matri& equation, and "ice-"ersa#

c) understand the cases that may arise concerning the consistency or inconsistency of linear

simultaneous equations, relate them to the singularity or other%ise of the corresponding square

matri&, and sol"e consistent systems'

Section 1: Solving simultaneous equations in 3 unno!ns

(n *+, you need to be able to sol"e a set of simultaneous equations in"ol"ing un$no%ns' The

usual elimination method can be used'

irst you try to eliminate one of the un$no%ns from the equations, to be left %ith equations in un$no%ns'

Then you eliminate a second un$no%n to be left %ith one equation in one un$no%n'

"#ample: ol"e the equations:

x. y. /z 0 1

x. y. 2z0 +

/x.y. z0 +/

Solution: (n these equations it is probably simplest to eliminate y first:

x.z 0 +/ -0

2x3z0 1 -0

4e&t %e can eliminate 5:

1x0 + .0

6e can no% see that

x 0

rom equation, %e get:

z 0 +/ 3 7 0 8

rom equation, %e ha"e

y 0 +/ 3 + 3 +7 0 -+/

$eometrical interpretation: 9ach of the three original equations represents aplanein dimensional space' The three planes intersect at the point , -+/, 8)'

; ; ;

ometimes %e ha"e a system of t%o equations in"ol"ing un$no%ns' (n such situations %e

cannot find a unique solution, but %e can sol"e the equations by introducing aparameter' The

solutions to the equations are then e&pressed in terms of this parameter'

"#ample %: ol"e the equations:

x3y. /z0 2

x. y3 z0


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Solution: 6e introduce a parameter: letz = t' 4ote %e could ha"e lety = torx = tinstead)

Then, on rearrangement, the equations become:

x3y0 2 - /t

x. y0 . t

4e&t %e need to eliminate another "ariable, such as y:

1x0 + - 2t .0

o,2+

1 1x t=

6e can then find y:

y 0 x3 2 . /t from )7 +21 1

2 /y t t= +

i'e' ++1 1y t= +

o o"erall the solutions are: 2+

1 1

x t= , ++1 1

y t= + ,z = t'

$eometrical interpretation: The original t%o equations represent planes in three dimensional

space' The t%o planes intersect to form a line' Our solutions gi"e all the points on this line'

Occasionally %e need to sol"e a single equation in"ol"ing un$no%ns' (n such situations %e

ob"iously cannot find a unique solution, but %e can sol"e the equations by introducing t%o

parameters' The solution to the equation is then e&pressed in terms of these parameters'

"#ample 3: ol"e the equation

x3 /y.z0 1

Solution: 6e introduce parameters, such asy = tandz = u'

Then

x0 '2 . y3 ust need to eliminatezfrom

the top equations)'


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1-1 .onsistent and inconsistent equations

?ll the equations that %e ha"e considered so far in this unit can be sol"ed they ha"e all been

consistentequations)' ome systems of equations cannot be sol"ed because the equations are

contradictory 3 such equations are called inconsistent'

"#ample: uppose %e tried to sol"e the equations

x3 y.z0 /

x. y3 z0

x. 8y3 2z0 +

6e could try to eliminatexfirst:

-7y. +7z0 + - 0

-+y . 8z0 < - 0

(f %e no% eliminatez:< 0 + -

Clearly this is a contradiction'

The equations therefore ha"e no solution' The equations are inconsistent'

; ; ;

ometimes %e may be gi"en equations, such as

x. y.z0 1

x3 y. /z 0 +

2x3y. 2z 0 8

but one of the equations might be able to be e&pressed in terms of the other t%o here the third

equations is the sum of the top t%o equations)' The equations are then called linearly dependent

equationsand can then only be sol"ed by introducing aparameter'

"#ample %: ol"e the system of equations:

x3y0 +

x. z0 +

y . /z0

Solution:

6e ha"e equations in un$no%ns' 6e could start by eliminating 5 %hich is already not presentin the top equation):

x3y0 + 0

7x3 y0 -0

6e can no% try to eliminatey:

< 0 < -

This is a true, but unhelpful, equation@ This means that the equations must ha"e been linearly

dependent equations in fact 0 - )'

6e therefore introduce a parameter, such asx = t' Then from equation, %e gety 0 + - t'


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?lso from: z0 + 3 t i'e' z 0 7'2 3 +'2t'

Summary:

Ai"en a system of equations in un$no%ns, there are possible scenarios:

+) There is a uniquesolution#

) There are an infinitenumber of solutions#) There is nosolution,

(n cases +) and ) the equations are called consistent' (n case ) the equations are called

inconsistent'

/ored e#amination question &'anuary %((3 *+*,

Three simultaneous equations are

2 2 :+)

, 2 +< :,)

1 :)

x y z

x y z

x y az

=

+ =

+ + =

%here ais a constant'a) or aB -, find the unique solution of the equations'

b) (n the case %hen a0 - find the general solution of the equations#

Solution:

a) (t is easiest to start by eliminatingy

2x3 +


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Section %: 0eterminants and inverses of 3 by 3 matrices

%-1 0eterminants of 3 by 3 matrices

The determinant of a general by matri& is defined as follo%s:

this is thethis is the this is thematri& lmatri& left matri& left

%hen the ro% %hen the ro%

and column and column

containing containing

are deleted are deleted

det

a b

a b ce f d f d e

d e f a b ch i g i g h

g h i

= + 1 2 3 1 2 3 {

eft

%hen the ro%

and column

containing

are deleted

c

4ote:e f

h istands for the determinant of the by matri&

e f

h i

'

"#ample: ind

+

det + 2

+ / ust the signs of e"ery other element starting %ith the second entry):

++ 7 1

+ +

/ 2

Step 2: Ta$e the transpose and di"ide by the determinant:

++ + /+

7 D

1 + 2

This is the in"erse matri&'

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