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INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
2007 Pearson Education Asia
Chapter 6 Chapter 6 Matrix AlgebraMatrix Algebra
2007 Pearson Education Asia
INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
1. Applications and More Algebra
2. Functions and Graphs
3. Lines, Parabolas, and Systems
4. Exponential and Logarithmic Functions
5. Mathematics of Finance
6. Matrix Algebra
7. Linear Programming
8. Introduction to Probability and Statistics
2007 Pearson Education Asia
9. Additional Topics in Probability
10. Limits and Continuity
11. Differentiation
12. Additional Differentiation Topics
13. Curve Sketching
14. Integration
15. Methods and Applications of Integration
16. Continuous Random Variables
17. Multivariable Calculus
INTRODUCTORY MATHEMATICAL ANALYSIS
2007 Pearson Education Asia
• Concept of a matrix.
• Special types of matrices.
• Matrix addition and scalar multiplication operations.
• Express a system as a single matrix equation using matrix multiplication.
• Matrix reduction to solve a linear system.
• Theory of homogeneous systems.
• Inverse matrix.
• Use a matrix to analyze the production of sectors of an economy.
Chapter 6: Matrix Algebra
Chapter ObjectivesChapter Objectives
2007 Pearson Education Asia
Matrices
Matrix Addition and Scalar Multiplication
Matrix Multiplication
Solving Systems by Reducing Matrices
Solving Systems by Reducing Matrices (continued)
Inverses
Leontief’s Input—Output Analysis
6.1)
6.2)
6.3)
6.4)
Chapter 6: Matrix Algebra
Chapter OutlineChapter Outline
6.5)
6.6)
6.7)
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.1 Matrices6.1 Matrices• A matrix consisting of m horizontal rows and n
vertical columns is called an m×n matrix or a matrix of size m×n.
• For the entry aij, we call i the row subscript and j the column subscript.
mnmm
n
n
aaa
aaa
aaa
...
......
......
......
...
...
21
21221
11211
2007 Pearson Education Asia
a. The matrix has size .
b. The matrix has size .
c. The matrix has size .
d. The matrix has size .
Chapter 6: Matrix Algebra
6.1 Matrices
Example 1 – Size of a Matrix
021 31
49
15
61
23
7 11
11126
865119
42731
53
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.1 Matrices
Example 3 – Constructing Matrices
Equality of Matrices
• Matrices A = [aij ] and B = [bij] are equal if they have the same size and aij = bij for each i and j.
Transpose of a Matrix
• A transpose matrix is denoted by AT.
If , find .
Solution:
Observe that .
654
321A
63
52
41TA
AATT
TA
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Chapter 6: Matrix Algebra
6.2 Matrix Addition and Scalar Multiplication6.2 Matrix Addition and Scalar Multiplication
Example 1 – Matrix Addition
Matrix Addition
• Sum A + B is the m × n matrix obtained by adding corresponding entries of A and B.
a.
b. is impossible as matrices are not of the same
size.
68
83
08
0635
4463
2271
03
46
27
63
52
41
1
2
43
21
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.2 Matrix Addition and Scalar Multiplication
Example 3 – Demand Vectors for an Economy
Demand for the consumers is
For the industries is
What is the total demand for consumers and the industries?
Solution:
Total:
1264 1170 523 321 DDD
0530 8020 410 SEC DDD
182571264 1170523321 DDD
1265005308020410 SEC DDD
3031571265018257
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.2 Matrix Addition and Scalar Multiplication
Scalar Multiplication
• Properties of Scalar Multiplication:
Subtraction of Matrices
• Property of subtraction is AA 1
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.2 Matrix Addition and Scalar Multiplication
Example 5 – Matrix Subtraction
a.
b.
13
08
84
3203
1144
2662
30
14
26
23
14
62
52
80
42
66
10
262BAT
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.3 Matrix Multiplication6.3 Matrix Multiplication
Example 1 – Sizes of Matrices and Their Product
• AB is the m× p matrix C whose entry cij is given by
A = 3 × 5 matrix
B = 5 × 3 matrix
AB = 3 × 3 matrix but BA = 5 × 5 matrix.
C = 3 × 5 matrixD = 7 × 3 matrixCD = undefined but DC = 7 × 5 matrix.
njinji
n
kjikjikij babababac
...221
11
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.3 Matrix Multiplication
Example 3 – Matrix Products
a.
b.
c.
d.
32
6
5
4
321
183
122
61
61
3
2
1
1047
0110
11316
212
312
201
401
122
031
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aa
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Chapter 6: Matrix Algebra
6.3 Matrix Multiplication
Example 5 – Cost Vector
Given the price and the quantities, calculate the total cost.
Solution:
The cost vector is
432P
C of units
B of units
Aof units
11
5
7
Q
73
11
5
7
432
PQ
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Chapter 6: Matrix Algebra
6.3 Matrix Multiplication
Example 7 – Associative Property
If
compute ABC in two ways.
Solution 1: Solution 2:
Note that A(BC) = (AB)C.
11
20
01
211
103
43
21CBA
196
94
43
12
43
21
11
20
01
211
103
43
21BCA
196
94
11
20
01
1145
521
11
20
01
211
103
43
21CAB
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.3 Matrix Multiplication
Example 9 – Raw Materials and Cost
Find QRC when
Solution:
975Q
1358256
21912187
17716205
R
1500
150
800
1200
2500
C
71650
81550
75850
1500
150
800
1200
2500
1358256
21912187
17716205
RC
900,809,1
71650
81550
75850
1275
RCQQRC
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.3 Matrix Multiplication
Example 11 – Matrix Operations Involving I and O
If
compute each of the following.
Solution:
00
00
10
01
41
23
103
101
51
52
OIBA
31
22
41
23
10
01 a.
AI
63
63
20
02
41
233
10
012
41
23323 b. IA
OAO
00
00
41
23 c.
IAB
10
01
41
23 d.
103
101
51
52
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Chapter 6: Matrix Algebra
6.3 Matrix Multiplication
Example 13 – Matrix Form of a System Using Matrix Multiplication
Write the system
in matrix form by using matrix multiplication.
Solution:
If
then the single matrix equation is
738
452
21
21
xx
xx
7
4
38
52
2
1 Bx
xXA
7
4
38
52
2
1
x
x
BAX
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Chapter 6: Matrix Algebra
6.4 Solving Systems by Reducing Matrices6.4 Solving Systems by Reducing Matrices
Elementary Row Operations
1. Interchanging two rows of a matrix
2. Multiplying a row of a matrix by a nonzero number
3. Adding a multiple of one row of a matrix to a different row of that matrix
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.4 Solving Systems by Reducing Matrices6.4 Solving Systems by Reducing Matrices
Properties of a Reduced Matrix
• All zero-rows at the bottom.
• For each nonzero-row, leading entry is 1 and the rest zeros.
• Leading entry in each row is to the right of the leading entry in any row above it.
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.4 Solving Systems by Reducing Matrices
Example 1 – Reduced Matrices
For each of the following matrices, determine whether it is reduced or not reduced.
Solution:
a. Not reduced b. Reduced
c. Not reduced d. Reduced
e. Not reduced f. Reduced
0000
2100
3010
f.
010
000
001
e. 000
000d.
01
10c.
010
001b.
30
01 a.
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Chapter 6: Matrix Algebra
6.4 Solving Systems by Reducing Matrices
Example 3 – Solving a System by Reduction
By using matrix reduction, solve the system
Solution:Reducing the augmented coefficient matrix of the system,
We have
1
5
1
11
12
32
1
52
132
yx
yx
yx
0
3
4
00
10
01
3
4
y
x
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.4 Solving Systems by Reducing Matrices
Example 5 – Parametric Form of a Solution
Using matrix reduction, solve
Solution:Reducing the matrix of the system,
We have and x4 takes on any real value.
9
2
10
6303
1210
6232
9633
22
06232
431
432
4321
xxx
xxx
xxxx
1
0
4
100
0010
001
21
25
421
3
2
425
1
1
0
4
xx
x
xx
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.5 Solving Systems by Reducing Matrices 6.5 Solving Systems by Reducing Matrices (continued)(continued)
Example 1 – Two-Parameter Family of Solutions
Using matrix reduction, solve
Solution:
The matrix is reduced to
The solution is
32
143
3552
4321
4321
4321
xxxx
xxxx
xxxx
0
2
1
0000
1210
3101
sx
rx
srx
srx
4
3
2
1
22
31
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.5 Solving Systems by Reducing Matrices (Continue)
• The system
is called a homogeneous system if c1 = c2 = … = cm = 0.
• The system is non-homogeneous if at least one of the c’s is not equal to 0.
mnmnmm
nn
cxaxaxa
cxaxaxa
...
.
.
.
.
...
2211
11212111
Concept for number of solutions:
1. k < n infinite solutions
2. k = n unique solution
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.5 Solving Systems by Reducing Matrices (Continue)
Example 3 – Number of Solutions of a Homogeneous System
Determine whether the system has a unique solution or infinitely many solutions.
Solution:
2 equations (k), homogeneous system, 3 unknowns (n).
The system has infinitely many solutions.
0422
02
zyx
zyx
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Chapter 6: Matrix Algebra
6.6 Inverses6.6 Inverses
Example 1 – Inverse of a Matrix
• When matrix CA = I, C is an inverse of A and A is invertible.
Let and . Determine whether C is
an inverse of A.
Solution:
Thus, matrix C is an inverse of A.
73
21A
13
27C
ICA
10
01
73
21
13
27
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.6 Inverses
Example 3 – Determining the Invertibility of a Matrix
Determine if is invertible.
Solution: We have
Matrix A is invertible where
Method to Find the Inverse of a Matrix
• When matrix is reduced, ,
- If R = I, A is invertible and A−1 = B.
- If R I, A is not invertible.
BRIA
22
01A
10
01
22
01IA BI
2
11
01
10
01
21
1
1
01A
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.6 Inverses
Example 5 – Using the Inverse to Solve a System
Solve the system by finding the inverse of the coefficient matrix.
Solution:
We have
For inverse,
The solution is given by X = A−1B:
1102
2 24
1 2
321
321
31
xxx
xxx
xx
1021
124
201
A
115
4
229
29
2411A
4
17
7
1
2
1
115
4
229
29
241
3
2
1
x
x
x
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.7 Leontief’s Input-Output Analysis6.7 Leontief’s Input-Output Analysis
Example 1 – Input-Output Analysis
• Entries are called input–output coefficients.• Use matrices to show inputs and outputs.
Given the input–output matrix,
suppose final demand changes to be 77 for A, 154 for B, and 231 for C. Find the output matrix for the economy. (The entries are in millions of dollars.)
2007 Pearson Education Asia
Chapter 6: Matrix Algebra
6.7 Leontief’s Input-Output Analysis
Example 1 – Input-Output Analysis
Solution:
Divide entries by the total value of output to get A:
Final-demand matrix:
Output matrix is
231
154
77
D
495
380
5.6921DAIX