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Singapore University of
Technology & Design
MATH 10.004Elimination & Augmented Form
Cohort 2
Meyer, Sections 1.2-1.3
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Elimination Matrices Elimination with Matrices Summary
LEARNING OBJECTIVES
After this cohort you will be able to ...
solve system of linear equations using elimination and back
substitution.
express systems in terms of matrices and perform basic
operations with matrices.
solve system of linear equations using augmented matrices.
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Elimination Matrices Elimination with Matrices Summary
ELIMINATION
Purpose is to provide a systematic way to solve systems of linearequations.
Commonly credited to Carl Friedrich Gauss, but first appearance
was in The Nine Chapters on the Mathematical Arts, an early
Chinese mathematics book composed by several authors and
completed around the 1st century.
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Elimination Matrices Elimination with Matrices Summary
ELIMINATION
Our aim is to simplify equations by performing the following types
ofrow operations:
(I) Exchange two equations.
(II) Multiply an equation by a non-zero constant.
(III) Add a multiple of one equation to another equation.
In particular, we want to eliminate variables to allow for easier
subsitution.
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EXAMPLE
Consider the following linear system
2x1` x2` x3 54x16x2 2
2x1`7x2`2x3 9
Well solve it via a series of elimination steps on the next slide.
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Elimination Matrices Elimination with Matrices Summary
Original system:
2x1` x2` x3 5 (1)4x16x2 2 (2)
2x1`7x2`2x3 9 (3)
We proceed byforward
elimination:
(2)-2(1)(2):
2x1` x2` x3 5
8x22x3 12
2x1`7x2`2x3 9
(3)+(1)(3):
2x1` x2` x3 5
8x22x3 12
8x2`3x3 14
(3)+(2)(3):
2x1` x2` x3 5
8x22x3 12
1x3 2
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Eli i i M i Eli i i i h M i S
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We now have the upper triangular system
2x1` x2` x3 5
8x2 2x3 12
1x3 2
which we may then solve viaback substitution:
1. From the third equation we havex32.
2. Pluggingx32 into the second equation, we then have
x2 14x3` 32 12` 32 1.
3. Plugging bothx32 andx21 into the first equation gives usx1
12
px2`x3q ` 52
1.
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Eli i ti M t i Eli i ti ith M t i S
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GAUSS-JORDAN METHOD
TheGauss-Jordan methodsimplifies back substitution.
2x1` x2` x3 5
8x2 2x3 12
1x3 2
2x1` x2 3
x2 1
x32
2x1` x2` 3
8x2 8
x3 2
x1 1
x2 1
x32
The leading coefficient in each equation is a 1.
Every coefficient of the same unknown above these leading
terms is zero.8
Elimination Matrices Elimination with Matrices Summary
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ACTIVITY 1: DOCKING A SPACE POD (15 MINUTES)
You find yourself in deep space piloting a small space pod that you
would like to dock to the mother ship.
You are currently stationary and your navigation tools define an
px, y, zqcoordinate system relative to your current position. Thedocking location is 4, 10, and 17 meters away (in x,y, andz).
You have 3 thrusters at your control. For each second you fire each
thruster the pod will move, in thex,y, andzdirections:
Thruster A: 1, 2, and 3 meters,
Thruster B: 1, 3, and 6 meters, Thruster C: 2, 6, and 10 meters.
Assuming a simple additive model of the interaction of the thrusters,
use elimination to find how many seconds each thruster needs to be
fired to move the pod to the dock.9
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Elimination Matrices Elimination with Matrices Summary
ACTIVITY 1: DOCKING A SPACE POD
Lets1,s2, ands3 be the time that each of the thrusters is fired. Using
matrix notation, the problem to solve may be written as:
s1`s2`2s3 4
2s1`3s2`6s310
3s1`6s2`10s317or after elimination:
1. p2q 2p1q p2q
2. p3q 3p1q p3q
3. p3q 3p2q p3q
s1`s2`2s3 4
s2`
2s3
2
2s3 1
which through back substitution then gives us the solution s12,s21, ands31{2.
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Elimination Matrices Elimination with Matrices Summary
MATRICES
Amatrixis a rectangular array of scalars.
If the matrix hasmrows andncolumns, we say that the size of
the matrix ismn.
The matrix is square ifmn.
The scalar in theith row andjth column is called thepi,jq-entry ofthe matrix.
Am n
a11 . . . a1j . . . a1n...
......
ai1 . . . aij . . . ain...
......
am1 . . . amj . . . amn
fiffiffiffiffiffiffifl
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SUM OF MATRICES
Matrices of the same dimensions can be added.
IfA
a11 . . . a1n
..
.
..
.am1 . . . amn
fiffifl
andB
b11 . . . b1n
..
.
..
.bm1 . . . bmn
fiffifl
,
then
A`Ba11`b11 . . . a1n` b1n
.
..
.
..am1`bm1 . . . amn` bmn
fiffifl .
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PRODUCT OF A MATRIX BY A SCALAR
Matrices can be multiplied by a scalar. If A
a11 . . . a1n...
...
am1 . . . amn
fiffifl
and P R, then
A
a11 . . . a1n..
.
..
.am1 . . . amn
fiffifl .
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PROPERTIES OF MATRIX OPERATIONS
Matrices obey the following laws related to addition:
A`BB`A (the commutative law)
cpA`Bq cA`cB(the distributive law)
A` pB`Cq pA`Bq `CA`B`C(the associative law)
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TRANSPOSE
Given a matrixA, itstransposeAT is obtained by interchanging the
role of rows and columns. For instance:
A24
1 2 7 03 2 1 6
AT
42
1 3
2 27 1
0 6
fiffiffifl .
pAT
qT
A pABqT AT BT
pAqT AT
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VECTORS AS MATRICES
Arow vectorof sizenis a 1nmatrix.
Acolumn vectorof sizemis anm1 matrix.
Ifvis a column vector, then vT is a row vector.
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PRODUCT OF MATRICES
Matrices can be multiplied if the number of columns of the first matrix
is equal to the number of rows of the second. If
Am n
a11 . . . a1n
......
am1 . . . amn
fi
ffifland B
n k
b11 . . . b1k
......
bn1 . . . bnk
fi
ffifl,then
Cm k
AB
is such that its genericpi,jq-entry has the form
cijn
h1
aihbhjAi B jApi, :q Bp:,jq.
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SIMPLE MATRIX MULTIPLICATION
For column vectorsu,v,wandy:
vTwv1 v2 v3
w1w2w3
fifl v1w1`v2w2`v3w3
uT
vT
w
u1 u2 u3
v1 v2 v3
w1w2w3
fifl
u1w1`u2w2`u3w3v1w1`v2w2`v3w3
vT
w y
v1 v2 v3
w1 y1w2 y2w3 y3
fifl
v1w1`v2w2`v3w3 v1y1`v2y2`v3y318
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SYSTEM OF LINEAR EQUATIONS
Alinear systemof mequations innunknowns is written as
a11x1`a12x2` a1nxnb1
a21x1`a22x2` a2nxnb2...
am1x1`am2x2` amnxnbm.
which is a mathematical way of expressingmlinear equality
constraints that thenvariablesxi, iP t1, . . . , nu, need to satisfy.
In matrix notation, we can write this compactly as
Axb,
whereA P Rm n,xP Rn, andbP Rm.19
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MATRIX FORM
We can rewrite the system:
2x1` x2` x3 5 (4)
4x16x2 2 (5)
2x1`7x2`2x3 9 (6)
as
Axb,
where
A
2 1 14 6 0
2 7 2
fifl , x
x1x2x3
fifl , b
52
9
fifl .
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ACTIVITY 2: MATRICES FOR DATA MANAGEMENT (20 MINUTES)
Create a matrixSthat records the day-end sales formstoresselling the samen items.
Price information will be stored in a n 1 matrix (vector) calledp.
What does theith
row ofStell you? And thejth
column?
What does thesijelement in the matrix tell you?
Find matrix operations to calculate:
A vector containing number of itemjs sold at each store. A vector containing the total revenue of each store.
The total revenue from all stores.
The total revenue of store i.
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ACTIVITY 2: MATRICES FOR DATA MANAGEMENT
S
s11 . . . s1n
......
sm1 . . . smn
fiffifl and p
p1
p2...
pn
fiffiffiffifl
Theith row gives the number of each item sold at store i.
Thejth column gives the number of item js sold at each of thestores.
The elementsijgives the number of itemjs sold at storei.
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ACTIVITY 2: MATRICES FOR DATA MANAGEMENT
A vector containing number of item js sold at each store:
Sej
s11 . . . s1j . . . s1n...
......
sm1 . . . smj . . . smn
fiffifl
0...
1..
.0
fiffiffiffiffiffiffifl
s1j...
smj
fiffifl
A vector containing the total revenue of each store:
Sp
s11 . . . s1n... ...sm1 . . . smn
fiffiflp1...pn
fiffifl rev1...
revm
fiffifl ,
where revkis the revenue made from all items at storek.
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ACTIVITY 2: MATRICES FOR DATA MANAGEMENT
The total revenue from all stores:
1TmSp
1 . . . 1
s11 . . . s1n...
...
sm1 . . . smn
fiffifl
p1...
pn
fiffifl
m
k1
revk.
The total revenue of storei:
eTiSp
0 . . . 1 . . . 0
s11 . . . s1n...
..
.si1 . . . sin
......
sm1 . . . smn
fiffiffiffiffiffiffiflp1...pn
fiffifl revi.
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ACTIVITY 2B (5 MINUTES)
Enter these commands into SCILAB, MATLAB, etc and discuss their
meaning:
clear all;
sales=[500 520 128 58; 850 600 54 32]
prices=[1.55 2.35 1.5 3.5]
revshop=sales*prices
revtot=[1 1]*revshop
revshopfruit=sales(:,1:3)*prices(1:3)revtotfruit=[1 1]*revshopfruit
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ACTIVITY 3(10 MINUTES)
Consider an electric circuit which takes input voltageVin and currentIinand produces output voltageVout and currentIout.
For a series circuit:
V2V1I1R1 andI2I1.
For a parallel circuit:
V3V2 andI3I2V2{R2.
Define a (transfer) matrixA such that:
V2
I2
A1
V1
I1
,
V3
I3
A2
V2
I2
,
V3
I3
A3
V1
I1
for the series circuit, parallel circuit, and combined circuit, respectively.
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ACTIVITY 3
According to Ohms law and Kirchhoffs circuit laws:
For the series circuit,V2V1I1R1,I2I1:
The transfer matrix of a series circuit is A1
1 R10 1
.
For the parallel circuit,V3V2,I3I2V2{R2:
The transfer matrix of a parallel circuit is A2
1 0
1{R2 1
.
We can use matrix multiplication to find the total transfer function:
V3
I3
A2
V2
I2
A2A1
loomoonA3V1
I1
, A3
1 R1
1{R2 1`R1{R2
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AUGMENTED FORM
Given the equationAxb
2 1 1
4 6 02 7 2
fi
fl
x1
x2
x3
fi
fl
5
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fi
fl,
we callA b
theaugmented matrixof the system.
A b
2 1 1 54 6 0 2
2 7 2 9
fifl
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Original system:
2x1` x2` x3 5 (1)
4x16x2 2 (2)
2x1`7x2`2x3 9 (3)
We proceed byforward
elimination:
(2)-2(1)(2):
2x1` x2` x3 5
8x22x3 12
2x1`7x2`2x3 9
(3)+(1)(3):
2x1` x2` x3 58x22x3 12
8x2`3x3 14
(3)+(2)(3):
2x1` x2` x3 5
8x22x3 12
1x3 2
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Original system:
2 1 1 54 6 0 22 7 2 9
fifl
We proceed byforward
elimination:
(2)-2(1)(2):
2 1 1 50 8 2 12
2 7 2 9
fifl
(3)+(1)(3):
2 1 1 50 8 2 12
0 8 3 14
fifl
(3)+(2)(3):
2 1 1 5
0 8 2 120 0 1 2fifl
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ELIMINATION WITH MATRICES
Using matrix notation, we started with the original system:
Ax
2 1 14 6 0
2 7 2
fifl x
52
9
fifl b,
and through a series of elementary row operations transformed it to
theequivalent system:
Ux
2 1 10 8 2
0 0 1
fifl x
512
2
fifl c.
The Gauss-Jordan method on the augmented matrix works the sameway as before:
Rx
1 0 0
0 1 0
0 0 1
fi
flx
1
1
2
fi
fld.
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ACTIVITY 4: AUGMENTED MATRICES (15 MINUTES)
Solve Activity 1: Docking a Space Pod using augmented matrices.
Swap rows 2 3 and solve through elimination again.
Are the final triangular systems the same?
Are the solutions the same?
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ACTIVITY 4: AUGMENTED MATRICES
Lets1,s2, ands3 be the time that each of the thrusters is fired. Using
matrix notation, the problem to solve may be written as:1 1 2 42 3 6 10
3 6 10 17
fifl
1 1 2 40 1 2 2
0 0 2 1
fifl
or after switching rows 2 3:
1 1 2 4
3 6 10 17
2 3 6 10
fi
fl
1 1 2 4
0 3 4 5
0 0 23
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fi
flElimination is not unique!
Either way substitution then gives us the same solution:
ps1, s2, s3q p2, 1, 1{2q.
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ACTIVITY 5: PRACTICE (10 MINUTES)
Using elimination on augmented matrices solve the following system
of linear equations.
x1`2x2` 8x37x4 2
3x1`2x2`12x35x4 6
x1` x2` x35x4 10
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ACTIVITY 5: PRACTICE
1 2 8 7 23 2 12 5 6
1 1 1 5 10
fifl
p2q 3p1q p3q ` p1q
1 2 8 7 20 4 12 16 120 3 9 12 12
fi
fl 3p2q 4p3q
1 2 8 7 20 12 36 48 36
0 12 36 48 48
fifl
p3q ` p2q
1 2 8 7 20 12 36 48 36
0 0 0 0 12
fifl
This system is inconsistent,
therefore it has no solution!
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SUMMARY
Elimination provides a systematic way to solve systems of linear
equations.
Systems of linear equations can be represented by a matrix
equation.
Basic definitions of matrices and their properties.
Augmented form is a compact way to solve linear systems withelimination.
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