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MATH ZC 161Engineering Mathematics-I
Lecture-1By
Dr. Deepmala Agarwal
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Learning Objectives
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2
To understand :Operations on matricesSpecial types of matrices
Use of matrices to study system of linear equationsRow echelon and row reduced echelon formsGaussian and Gauss-J ordan elimination as application.Rank using row reduced echelon form.
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2nd row
2nd column
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An m x n matrix has m rows and nColumns, as shown here.
nm)(a
:wayeAlternativ
ij
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4
Row Vector
vectorrowcalledis
)(
matrix1A
21 naaaA
n
=
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5
Column Vector
vector.columncalledis
2
1
=
ma
a
a
A
matrix1A n
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6
Square Matrix
.33orderofmatrixsquareais
312
163
745
:exampleFor
=B
A matrix in which number of rowsis same as number of columns
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Matrix Addition and Subtraction
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size)Samehave(All)(
,)(
thensize),(sameandIf
nmijij
nmijij
nmijnmij
baBA
baBA
)(bB)(aA
=
+=+
==
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Addition
A = a11 a12
a21 a22
B = b11 b12
b21 b22
C = a11 +b11 a12 +b12a21 +b21 a 22 +b22
If
and
then
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9
Matrix Addition Example
A + B = 3 4
5 6
+
1 2
3 4
=
4 6
8 10
= C
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1127/7/2011 MATH ZC161 EngineeringMathematics I
scalar.acalledis
.912
15633435323
34523
:exampleFor
.)(
thennumber,complexorrealaisIf
k
kakA
)(aAk
nmij
nmij
=
=
=
=
:tionMultiplicaScalar
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12
Matrix Multiplication
Matrices A and B have these dimensions:
[r x c] and [s x d]
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Matrix Multiplication
Matrices A and B can be multiplied if:
[r x c] and [s x d]
c = s
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Matrix Multiplication
The resulting matrix will have the dimensions:
[r x c] and [s x d]
r x d
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Zero Matrix
0 0...... 0
0 0...... 0
0 0...... 00
. . .
. . .0 0...... 0
=
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Triangular Matrix
1 2 3 4 1 0 0 0
0 5 6 8 1 6 0 0
0 0 11 5 7 9 3 0
0 0 0 9 2 3 4 6
Upper TriangularMatrix
Lower TriangularMatrix
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Symmetric Matrix
A matrix An n is said to be symmetric if
AT
= A
Ex: A=
1 2 7
2 5 6
7 6 4
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Systems of Linear Equations
A system ofn linear equations in n variables,
can be expressed as a matrix equationAx =b:
Ifb =0, then system is homogeneous; otherwise it is
nonhomogeneous.
=
nnnnnn
n
n
b
b
b
x
x
x
aaa
aaa
aaa
2
1
2
1
,2,1,
,22,21,2
,12,11,1
,,22,11,
2,222,211,2
1,122,111,1
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
=+++
=+++
=+++
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Matrix in Row Echelon form :
1.1st non-zero entry of any non-zero row is 1.2.In consecutive non-zero rows, 1st entry 1 in the lowerrow appears to the right of 1st entry 1 of the upper row.3. Rows containing all 0s are at the bottom of the matrix.
Matrix in Reduced Row Echelon form :
This is matrix which is in row echelon form with further
property that if a column contains a 1st
entry 1 of anyrow then all other entries of that column are 0.
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Elimination methods :
We have two ways :
1. Gaussian elimination : Use elementary row operations toconvert the augmented matrix to a row echelon form.Solve the resulting system corresponding to row echelonform using back substitution.These are solutions are the solutions of the original system.
2. Gauss-Jordan Elimination : Use elementary row operations toconvert the augmented matrix to the reduced row echelon form.Advantage :No back substitution required to solve this system.
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Remark : In Gaussian elimination, the sequence ofrow operations required may not be unique, also
the result, a row echelon form, may not be unique.
In Gauss-J ordan elimination, the sequence of row
operations may not be unique, bur reduced rowechelon form is unique.
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.113217532
3111
issystemtheofmatrixaugmentedThe.1132
7532
3exist.notdoessolutionshow the
orsystemthesolvetoneliminatioJordan-Gauss
orneliminatioGaussianeitherUse:
321
321
321
=+
=++
=
xxx
xxx
xxx
366)Ex.5(p
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lli i iG i
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form.echelonrowaisThis.
1100
8410
3111
272700
8410
3111
13750
8410
3111
13750
84103111
8410
137503111
11321
13750
3111
11321
7532
3111
form.echelonrow
amatrix toaugmentedisconvert thtooperations
rowelementaryuseWe:
27
5
2
3
232
2313
12
R
RRR
RRR
RR
neliminatioGaussian
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also.systemoriginalsolnisThis.0giveseqn
1stinvariables2theseofvaluesthengSubstituti
.4getweeqn,2ndinthisngSubstituti.
givesequationlaston,substitutibackbysolveTo
.1
84
3
isformechelon
rowthistoingcorrespondsystemresultingThe
1
23
3
32
321
=
=
=
==
x
xx
x
xx
xxx
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form.echelonrow
reducedgettoneliminatioGaussianinobtained
formechelonreduced-rowtheonoperationsrow
elementaryuseWe:neliminatioJordan-Gauss
soln.(same)get thedirectlyweandrequirednotis
onsubstitutibackthatseeweneliminatioGaussianinas
systemtheFormingform.echelonrowreducedisThis
.
1100
4010
0001
1100
8410
0001
1100
8410
5501
1100
8410
3111
3231
21
45
R
++
+
RRRR
R
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J dGli i tiG iU366)9(E
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.
4111-
311-1
81-1-1
issystemtheofmatrixaugmentedThe
4
3
8
exist.notdoessolutionthe
showorequationslinearofsystemthesolvetoneliminatio
Jordan-GaussorneliminatioGaussianUse:
321
321
321
=++
=+
=
xxx
xxx
xxx
366)9(pEx
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form.echelonrowisThis10005/2-100
81-1-1
12000
5/2-10081-1-1
12000
5-20081-1-1
4111-
5-200
81-1-1
4111-
311-1
81-1-1
12
2
3
2
23
12
+
R
RRR
RR
Gaussian elimination : We use elementary row operations toconvert this augmented matrix to row echelon form.
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.102/5
8
isformechelonrow
thistoingcorrespondequationslinearofsystemThe
3
321
==
=
x
xxx
The system has no solution, as the last equation can never besatisfied.
Thus the original system is also inconsistent. (has no solution)
Try Gauss-J ordan elimination yourself.27/7/2011 42MATH ZC161 Engineering Math. I
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Example
.3,2
7,
2
5
solutiontrivialnonahas
020342
unknownsin threeequationstwoofsystemhomgeneousThe
321
321
321
===
=+
=+
xxx
xxx
xxx
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Rank using echelon form
The definition of the rank of a matrix is more involved andunsuitable for our purpose. For all practical reasons we canuse its characterization involving Reduced row echelon form.
Steps : To find rank of an m x n matrix AReduce A to the reduced row echelon formusing a sequence ofelementary row operations
Count the number ofnon-zero rows in it to get the rank of A
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Exercise.Try the following problems based on lectures 1On p 349: 38, 39, 40. On p 359: 8, 10 ,17 ,19. On p 364: 3, 4,6, 7, 8.
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