3. On the basis of the matrices given in Example 9, is the product BA defined? If so, calculate the product. In this case do we have AB = BA?
3. On the basis of the matrices given in Example 9, is the product BA defined? If so, calculate the product. In this case do we have AB = BA?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
Related questions
Question
100%
Solve Q3
see the attached files
![Ath Fundamental Methods of Mathem...
i = 1, 2, ..., m
l;= 1,2,
11
This last equation represents yet another way of stating the rule of multiplication for the
matrices defined above.
EXERCISE 4.2
0.
8 =
3
1. Given A =
and C =
find:
6 1
(C) 3A
(b) C - A
[2 87
2. Given A = 3 0,8 =
5 1
(0) A + B
(d) 4B + 2C
,and C =
3 8
(a) Is AB defined Calculate
UIS BC defined? Całculate BC. Is CB defined? If so, całculate CB. Is it true that BC= CB?
3. On the basis of the matrices given in Example 9, is the product BA defined? If so,
calculate the product. In this case do we have AB = BA?
4. Find the product matrices in the foltowing (in each case, append beneath eve X
BA? Why?
O 2 01[8 0
0 1
(c)
(0) 3 0 4
2 3 0 L3 5
4 -1
2 -7
6 5 -1
(b)
1 0
(d) [a b c] 0 2
1 4
2
4
5. In Example 7, if we arrange the quantities and prices as column vectors instead of row
vectors, is Q- P defined? Can we express the total purchase cost as Q P? As Q' - P? As
Q.P'?
6. Expand the following summation expressions:
(a) xi
2
( ax-
i=1
i-s
(c) bx,
Chapter 4 Lineur Models and Marrix Algebra 59
7. Rewrite the foltowing in E notation:
(0) x1 (X1 – 1) + 2x2{x2 – 1) + 3x3(x3 – 1)
(b) az{x3 + 2) + a3(X4 + 3) + C4(xs + 4)
1
1
(c) - +
(x + 0)
1
1
(d) 1+
x x2
(x # 0)
+
8. Show that the foliowing are true:
n-1
(a)
+ Xn+1 = E X;
(b) E ab, Y; = a
(c) E (x; + y,) = E x; +
4.3 Notes on Vector Operations
In Secs. 4.1 and 4.2, vectors are considered as a special type of matrix. As such, they qual-
ify for the appiication of all the algebraic operations discussed. Owing to their dimensional
nesuliarities k
addirional comment](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F854ac530-fc47-4f9f-9340-e5e86f181044%2F66730457-9af3-42a6-a3b5-76f17a6c30a2%2Fw4wiz8q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Ath Fundamental Methods of Mathem...
i = 1, 2, ..., m
l;= 1,2,
11
This last equation represents yet another way of stating the rule of multiplication for the
matrices defined above.
EXERCISE 4.2
0.
8 =
3
1. Given A =
and C =
find:
6 1
(C) 3A
(b) C - A
[2 87
2. Given A = 3 0,8 =
5 1
(0) A + B
(d) 4B + 2C
,and C =
3 8
(a) Is AB defined Calculate
UIS BC defined? Całculate BC. Is CB defined? If so, całculate CB. Is it true that BC= CB?
3. On the basis of the matrices given in Example 9, is the product BA defined? If so,
calculate the product. In this case do we have AB = BA?
4. Find the product matrices in the foltowing (in each case, append beneath eve X
BA? Why?
O 2 01[8 0
0 1
(c)
(0) 3 0 4
2 3 0 L3 5
4 -1
2 -7
6 5 -1
(b)
1 0
(d) [a b c] 0 2
1 4
2
4
5. In Example 7, if we arrange the quantities and prices as column vectors instead of row
vectors, is Q- P defined? Can we express the total purchase cost as Q P? As Q' - P? As
Q.P'?
6. Expand the following summation expressions:
(a) xi
2
( ax-
i=1
i-s
(c) bx,
Chapter 4 Lineur Models and Marrix Algebra 59
7. Rewrite the foltowing in E notation:
(0) x1 (X1 – 1) + 2x2{x2 – 1) + 3x3(x3 – 1)
(b) az{x3 + 2) + a3(X4 + 3) + C4(xs + 4)
1
1
(c) - +
(x + 0)
1
1
(d) 1+
x x2
(x # 0)
+
8. Show that the foliowing are true:
n-1
(a)
+ Xn+1 = E X;
(b) E ab, Y; = a
(c) E (x; + y,) = E x; +
4.3 Notes on Vector Operations
In Secs. 4.1 and 4.2, vectors are considered as a special type of matrix. As such, they qual-
ify for the appiication of all the algebraic operations discussed. Owing to their dimensional
nesuliarities k
addirional comment
![6:53 1
4G
K Ath Fundamental Methods of Mathem...
of A and B are other than those illustratcu in Fig. 4.1, tie ony prerequisnE I5 uhat uie Coi-
formability condition be met.
Given
Example 8
[1 3
A =2 8
4 0
and
B =
(2x1
(3x2)
Chapter 4 Linear Models und Matrix Algebru 55
find AB. The product AB is indeed defined because A has two columns and B has two rows.
Their product matrix should be 3 × 1, a column vector:
[1(5) + 3(9) 7
AB =| 2(5) + 8(9)
4(5) + 0(9) ]
T 32
82
20
Given
Example 9
-1
A =
(33)
0 3
0 2
and
B =
(3x 3)
-1
find AB. The same rule of multiplication now yields a very special product matrix:
[0+1+0 -}-} + - o -
[10 0
AB =| 0+0+0 -}+0+ % + 0 - =|0 1 0
L0 0 1
0+0+0 -+0+ + 0 -
This last matrix-a square matrix with 1s in its principai diagona! (the diagonal running from
northwest to southeast) and Os everywhere else-exemplifies the important type of matrix
known as the identity matrix. This will be further discussed in Section 4.5.
Example 10 Let us now take the matrix A and the vector x as defined in (4.4) and find Ax. The product
matrix is a 3 x 1 column vector:
9.
3
Ax =| 1
6x1 + 3x2 + X3
X1 + 4x2 – 2x3
4x1 - x2 + 5x3
X1
4 -2
X2
4 -1
(3x3)
(3x1)
(3x1)
Note: The product on the right is a column vector, its corpulent appearance notwithstand-
ing! When we write Ax = d, therefore, we have
[6x1 +3x2 + x3
Xi +4x2 - 2x3
4x — *2 + 5х
22
%3D
12
10
which, according to the definition of matrix equality, is equivaient to the statement of the
entire equation system in (4.3).
Note that, to use the matrix notation Ax = d, it is necessary, because of the conforma-
bility condition, to arrange the variables x; into a cołumn vector, even though these vari-
ables are listed in a horizontal order in the original equation systern.
Example 11
The simple national-income model in two endogenous variables Y and C,
Y = C + lo + Go
C = a+ bY
56 Part Two Static (or Equilibrium) Analysis
can be rearranged into the standard format of (4.1) as follows:
Y -C = lo + Go
-bY +C = a
Hence the coeffitient marix A, the vector or vanaOTES X, alra thẹ vector of constants d are](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F854ac530-fc47-4f9f-9340-e5e86f181044%2F66730457-9af3-42a6-a3b5-76f17a6c30a2%2F8y05oz_processed.png&w=3840&q=75)
Transcribed Image Text:6:53 1
4G
K Ath Fundamental Methods of Mathem...
of A and B are other than those illustratcu in Fig. 4.1, tie ony prerequisnE I5 uhat uie Coi-
formability condition be met.
Given
Example 8
[1 3
A =2 8
4 0
and
B =
(2x1
(3x2)
Chapter 4 Linear Models und Matrix Algebru 55
find AB. The product AB is indeed defined because A has two columns and B has two rows.
Their product matrix should be 3 × 1, a column vector:
[1(5) + 3(9) 7
AB =| 2(5) + 8(9)
4(5) + 0(9) ]
T 32
82
20
Given
Example 9
-1
A =
(33)
0 3
0 2
and
B =
(3x 3)
-1
find AB. The same rule of multiplication now yields a very special product matrix:
[0+1+0 -}-} + - o -
[10 0
AB =| 0+0+0 -}+0+ % + 0 - =|0 1 0
L0 0 1
0+0+0 -+0+ + 0 -
This last matrix-a square matrix with 1s in its principai diagona! (the diagonal running from
northwest to southeast) and Os everywhere else-exemplifies the important type of matrix
known as the identity matrix. This will be further discussed in Section 4.5.
Example 10 Let us now take the matrix A and the vector x as defined in (4.4) and find Ax. The product
matrix is a 3 x 1 column vector:
9.
3
Ax =| 1
6x1 + 3x2 + X3
X1 + 4x2 – 2x3
4x1 - x2 + 5x3
X1
4 -2
X2
4 -1
(3x3)
(3x1)
(3x1)
Note: The product on the right is a column vector, its corpulent appearance notwithstand-
ing! When we write Ax = d, therefore, we have
[6x1 +3x2 + x3
Xi +4x2 - 2x3
4x — *2 + 5х
22
%3D
12
10
which, according to the definition of matrix equality, is equivaient to the statement of the
entire equation system in (4.3).
Note that, to use the matrix notation Ax = d, it is necessary, because of the conforma-
bility condition, to arrange the variables x; into a cołumn vector, even though these vari-
ables are listed in a horizontal order in the original equation systern.
Example 11
The simple national-income model in two endogenous variables Y and C,
Y = C + lo + Go
C = a+ bY
56 Part Two Static (or Equilibrium) Analysis
can be rearranged into the standard format of (4.1) as follows:
Y -C = lo + Go
-bY +C = a
Hence the coeffitient marix A, the vector or vanaOTES X, alra thẹ vector of constants d are
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