Post on 18-Dec-2015
Matrices & Systems of Linear Equations
Special Matrices
000
000
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Examples
MatrixZero
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8102
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912
601
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01,5
:Examples
MatrixnbynAn
MatrixSquare
Special Matrices
8000
0500
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0001
400
010
001
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01,5
:Examples
zeroesarediogonal
maintheonnotarethatentriesall
ifmatrixsquareA
MatrixDiagonal
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0100
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0001
100
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Examples
onesarediogonal
maintheonarethatentriesall
ifmatrixdiagonalisA
MatrixIdentity
Equality of Matrices
Two matrices are said
to be equal if they have
the same size and their
corresponding entries
are equal
45
21,
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21
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584
573
062
951
,
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equalmatricesfollowingtheAre
equalmatricesfollowingtheAre
Equality of Matrices
Use the given equality
to find x, y and z
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21
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z
yx
z
y
x
Matrix Addition and SubtractionExample (1)
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1296
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664524
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318
642
150
987
654
321
Matrix Addition and SubtractionExample (2)
671
012
231
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135201
318
642
150
987
654
321
Multiplication of a Matrix by a Scalar
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512
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150
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Example
Example
Matrix Multiplication(n by m) Matrix X (m by k) Matrix
The number of columns of the matrix on the left
= number of rows of the matrix on the right
The result is a (n by k) Matrix
Matrix Multiplication3x3 X 3x3
332211332211332211
332211332211332211
332211332211332211
333
222
111
321
321
321
zczczcycycycxcxcxc
zbzbzbybybybxbxbxb
zazazayayayaxaxaxa
zyx
zyx
zyx
ccc
bbb
aaa
Matrix Multiplication1x3 X 3x3→ 1x3
32211332211332211
333
222
111
321
azazayayayaxaxaxa
zyx
zyx
zyx
aaa
Example (1)
323
422
241
)2)(1()1)(1()0)(2()1)(1()1)(1()1)(2()1)(1()0)(1()2)(2(
)2)(2()1)(0()0)(0()1)(2()1)(0()1)(0()1)(2()0)(0()2)(0(
)2)(1()1(4)0(1)1)(1()1(4)1(1)1)(1()0(4)2(1
211
110
012
112
200
141
Example (2)(1X3) X (3X3) → 1X3
52411
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432
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310
432
Example (3)(3X1) X (1X2) → 3X2
00
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Example (4)
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Transpose of Matrix
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333
222
111
321
321
321
T
T
Example
cba
cba
cba
A
ccc
bbb
aaa
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Example
Properties of the Transpose
TTT
TT
ABAB
AA
)(.2
)(.1
Matrix ReductionDefinitions (1)
1. Zero Row: A row consisting entirely of zeros
2. Nonzero Row: A row having at least one nonzero entry
3. Leading Entry of a row: The first nonzero entry of a row.
Matrix ReductionDefinitions (2)
Reduced Matrix: A matrix satisfying the following:
1. All zero rows, if any, are at the bottom of the matrix
2. The leading entry of a row is 1
3. All other entries in the column in which the leading entry is located are zeros.
4. A leading entry in a row is to the right of a leading entry in any row above it.
Examples of Reduced Matrices
000
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010
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100
010
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Examples matrices that are not reduced
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columntheintheNotice
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Elementary Row Operations
1. Interchanging two rows
2. Replacing a row by a nonzero multiple of itself
3. Replacing a row by the sum of that row and a nonzero multiple of another row.
Interchanging Rows
225
320
1263
225
1263
32021 RR
Replacing a row by a nonzero multiple of itself
225
320
421
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320
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3
1R
Replacing a row by the sum of that row and a nonzero multiple of another row
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Augmented Matrix Representing a System of linear Equations
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Solution
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Example
Solving a System of Linear Equations by Reducing its Augmented Matrix
Using Row Operations
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Solution
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Solution of the System
1
2
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Thus
The Idea behind the Reduction Method
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equationslinearofsystemThe
Interchanging the First & the Second Row
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Multiplying the first Equation by 1/3
7
7
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320
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Subtracting from the Third Equation 5 times the First Equation
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7
1
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7
1
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Systems with infinitely many Solutions
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solutionnohaswhich
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21
31
Finding the Inverse of an nXn square Matrix A
1. Adjoin the In identity matrix to obtain the Augmented matrix [A| In ]
2. Reduce [A| In ] to [In | B ] if possible
Then
B = A-1
Example (1)
1
115
014
001
100
920
201
101
014
001
820
920
201
100
014
001
1021
920
201
100
010
001
1021
124
201
:
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201
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31
21
)(
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Solution
AFindALet
115
4
229
,
115
4
229
100
010
001
115
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229
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001
115
014
229
100
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29
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29
241
)9(
)2(
221
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13
A
Thus
R
RR
RR
Example (2)
inversenohasAThus
Solution
AFindALet
RR
RR
RR
111
012
001
000
430
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103
012
001
430
430
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Inverse MatrixThe formula for the inverse of a 2X2 Matrix
AAIAA
thatCheck
ac
bd
AA
ThenAIf
bcadA
dc
baA
Let
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1
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det
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Checking
AA
A
ALeT
Example
Using the Inverse Matrixto Solve System of Linear Equations
12
:
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1
2
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2()9)(1(
)15)(3
5()9(3
15
9
3
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isSolutionThe
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y
x
y
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y
x
yx
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formmatrixinwriting
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equationslinearofsystemfollowingtheSolve
Problem
101
124
113
212
123
112
:
122
223
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thatgivenIf
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equationslinearofsystemfollowingtheSolve
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:
,
2
1
2
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1
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3
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Thus
z
y
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Homework
6.1 Examples: 3 Exercises: 17 - 20
6.2 Examples: 1, 2, 4, 5 and 6 Exercises: odd numbered:1—17 and 25,29,35,37,39,41
6.3 Examples: 1, 2, 3, 4, 5, 7, 10, 11, 12 and 13. Exercises:19, 21, 23, 25, 27, 31, 33, 37, 51, 53, 57, 59, 61
6.5 Examples: 2, 3.a and 4. See given exercises.
6.6 Examples: 1 - 6 Exercises: odd numbered:1—15, 21, 27, 29, 35 and 37.
1.