Matrices. Special Matrices Matrix Addition and Subtraction Example.
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Transcript of Matrices. Special Matrices Matrix Addition and Subtraction Example.
Matrices
Special Matrices
000
000
000
,0
0
0000,00
00
:
3321
1422
OO
OO
Examples
MatrixZero
0214
4513
8102
8101
453
912
601
,25
01,5
:Examples
MatrixnXnAn
MatrixSquare
Special Matrices
8000
0500
0070
0001
400
010
001
,20
01,5
:Examples
zeroesarediogonal
maintheonnotarethatentriesall
ifmatrixsquareA
MatrixDiagonal
1000
0100
0010
0001
100
010
001
,10
01,1
:
44
332211
I
III
Examples
onesarediogonal
maintheonarethatentriesall
ifmatrixdiagonalisA
MatrixIdentity
Special Matrices
6000
1700
3960
4581
700
510
412
,30
12
:
44
3322
I
II
Examples
zerosarediogonal
mainthebelowarethatentriesall
ifmatrixsquareA
MatrixDiogonalUpper
6129
1731
0068
0004
758
016
002
,31
02
:
44
3322
I
II
Examples
zerosarediogonal
maintheabovearethatentriesall
ifmatrixdiagonalA
MatrixDiogonalLower
Matrix Addition and SubtractionExample
671
012
231
391887
664524
135201
318
642
150
987
654
321
Multiplication of a Matrix by a Scalar
21176
666
512
318
642
150
181614
12108
642
318
642
150
)1(
987
654
321
2
)2(
45535
30020
15105
)9(5)1(5)7(5
)6(5)0(5)4(5
)3(5)2(5)1(5
917
604
321
5
)1(
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
42411
)4(4)10(3)3(2)3(4)2(3)1(2)2(4)1(3)0(2
432
1021
310
432
Example (3)(3X1) X (1X2) → 3X2
00
76
3530
)7(0)6(0
)7(1)6(1
)7(5)6(5
76
0
1
5
Example (4)
2
2
37164
1110
043
540
311
121
540
311
121
540
311
121
Transpose of Matrix
863
752
041
870
654
321
)1(
333
222
111
321
321
321
T
T
Example
cba
cba
cba
A
ccc
bbb
aaa
A
30273
251214
9102
200
020
002
32273
251014
9104
100
010
001
2
24210
181512
963
863
752
041
2
870
654
321
3
870
654
321
)2(
3I
Example
T
870
654
321
870
654
321
)3(
TT
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 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
710
501
.3
000
010
001
.2
100
010
001
.1
Examples matrices that are not reduced
.
2
000
001
010
.2
12
100
060
001
.1
itaboverowtheinentry
leadingtheofrightthetonotisrowinentryleadingThe
notisrowinentryleadingThe
zerosarelocatedisitwhichincolumnthein
entriesotherallthebutisrowinentryleadingThe
matrixtheofottomtheatnotbutrowzeroaisRow
,12
000
210
031
.4
.2
010
000
001
.3
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
225
320
12631
3
1R
Replacing a row by the sum of that row and a nonzero multiple of another row
22120
320
421
225
320
421)5( 13 RR
Augmented Matrix Representing a System of linear Equations
7
3
7
225
1263
320
:
7225
31263
732
:
matrixaugmentedthebydrepresenteIs
zyx
zyx
zy
equationslinearofsystemThe
29
)(
3
4
10
01
9
4
20
01
9
5
20
21
6
5
43
21
:
6
5
43
21
:
Re
)1(
221
21
12
R
RR
RR
Solution
operations
rowbasicapplyingbymatrixaugmentedfollowing
theofrihgttheonmatrixtwobytwothetheduce
Example
Solving a System of Linear Equations by Reducing its Augmented Matrix
Using Row Operations
29
29
&4
:
4
10
01
:
6
5
43
21
:
:
:
643
52
:
)1(
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thatconcludeWe
matrixtheatarrivingmatrixthisreduceWe
operations
matrixaugmentedtheconstructWe
Solution
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yx
equationslinearofsystemfollowingtheSolve
Example
7
3
7
225
1263
320
:
7225
31263
732
:
)2(
matrixaugmenteditsreducingby
zyx
zyx
zy
equationslinearofsystemfollowingtheSolve
Example
Solution
2
8
22120
10
701
2
1
22120
10
421
2
7
1
22120
320
421
7
7
1
225
320
421
7
7
3
225
320
1263
7
3
7
225
1263
320
27
23
)2(
27
23
2
1
)5(3
1
212
131
21
RRR
RRR
RR
1
2
1
100
010
001
1
1
100
10
001
1
8
100
10
701
40
8
4000
10
701
2
8
22120
10
701
)(
27
23
7
27
23
40
1
27
23
12
27
23
323
2
313
23
RR
RRR
RR
Solution of the System
1
2
1
,
1
2
1
100
010
001
z
y
x
Thus
The Idea behind the Reduction Method
7
3
7
225
1263
320
:matrixaugmentedThe
7225
31263
732
:
zyx
zyx
zy
equationslinearofsystemThe
Interchanging the First & the Second Row
7
7
3
225
320
1263
7
3
7
225
1263
32021 RR
7225
732
31263
7225
31263
732
:)2()1(
zyx
zy
zyx
zyx
zyx
zy
EqandEqBetweem
placesthengtIntercangi
Multiplying the first Equation by 1/3
7
7
1
225
320
421
7
7
3
225
320
1263
13
1R
7225
732
142
7225
732
31263
zyx
zy
zyx
zyx
zy
zyx
Subtracting from the Third Equation 5 times the First Equation
2
7
1
22120
320
421
7
7
1
225
320
421
)5( 13 RR
22212
732
142
7225
732
142
zy
zy
zyx
zyx
zy
zyx
Subtracting from the First Equation 2 times the Second Equation
2
8
22120
10
701
2
1
22120
10
421
27
23
)2(
27
23
21 RR
222122
7
2
3
87
222122
7
2
3
142
zy
zy
zx
zy
zy
zyx
Adding to the Third Equation 12 times the Second Equation
40
8
4000
10
701
2
8
22120
10
701
27
23
12
27
23
23 RR
40402
7
2
3
87
222122
7
2
3
87
z
zy
zx
zy
zy
zx
Dividing the Third Equation by 40
1
8
100
10
701
40
8
4000
10
701
27
23
40
1
27
23
3R
12
7
2
3
87
40402
7
2
3
87
z
zy
zx
z
zy
zx
Adding to the First Equation 7 times the third Equation
1
1
100
10
001
1
8
100
10
701
27
23
7
27
23
31 RR
12
7
2
3
1
12
7
2
3
87
z
zy
x
z
zy
zx
Subtracting from the Second Equation 3/2 times the third Equation
1
2
1
12
7
2
3
1
z
y
x
z
zy
x
1
2
1
100
010
001
1
1
100
10
001
)(
27
23
323
2 RR