Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.1: Matrix Operations
Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...
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Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
Question
100%
#53 please
![0.6
21. 0.7
-0.3]
0.1
0.2 0.3
22. -0.3
0.2
43. A-
B--
-1
0.2
0.2
-0.9
0.5
0.5
0.
-4
6
44. A -0
B--2
4
23. 3
2
24. 3
2
4
4
3
4
5
5
5 5
Ce tcer e
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72
Chapter 2 Matrices
Solving a System of Equations Using an Inverse
In Exercises 45-48, use an inverse matrix to solve each
system of linear equations.
Singular Matrix In Exercises 55 and 56, find x such
that the matrix is singular.
.
55. A -[-
56. A -
46. (a) 2x - y- -3
2r +y= 7
45. (a) x+ 2y --
x- 2y - 3
Solving a Matrix Equation In Exercises 57 and 58,
find A.
(b) x+ 2y - 10
(b) 2x - y--I
x- 2y - -6
2x +y--3
57. (24)- -
58. (44)- -
47. (a) x, + 2x, + x,- 2
X + 2x, - - 4
X - 2x, + --2
(b) x, + 2x, + x,- I
X, + 2x, - x,- 3
X- 2x, + x, --3
48. (a) x, + X, - 2x, - 0
X- 2x, + x - 0
X,- X,- x,--1
(b) x, + x - 2x, - -I
,- 2x, + , - 2
Finding the Inverse of a Matrix In Exercises 59
and 60, show that the matrix is invertible and find its
inverse.
59, a - [-
[sec e tan e
sin e co f
|- cos 6
sin e 60. A
sin e
I
tan e sec e
Beam Deflection In Exercises 61 and 62, forces w.W
and w, (in pounds) act on a simply supported elastic
beam, resulting in deflections d, dz, and d, (in inches) in
the beam (see figure).
Solving a System of Equations Using an Inverse
In Exercises 49-52, use a software program or a
graphing utility to solve the system of linear equations
using an inverse matrix.
49. X, + 2x, - , + 3x, - --3
X - 3x, + x, + 2x, - X = -3
2r, + x + 1, - 3x, + x, = 6
X- + 2x, + -X= 2
2x, + -A + 2x, + X- -3
Use the matrix equation d- Fw, where
--E
d =
50. , + x - x, + 3x, - x, - 3
2x, + 1 + + + I = 4
X, + X - x + 2x,- x,- 3
2x, +x, + 4x, + x- x, - -1
3x, +x+ - 2, + x- 5
51. 2r, - 3x, + x - 2, + Xg - 4xg = 20
3x, + - 4x, + - + 2x,-- 16
4x, + - 3x, + 4, - + 2,- -12
-Sx, - + 4x, + 2x, - 5x, + 3x, -2
X+ - 3x,+ 4 - 3x, + X--15
3x, - X + 2x, - 3x, + 2r, - 6r, - 25
4x, - 2x, + 4x, + 2, - Sx, - - I
3x, + 6x, - Sx, - 6, + 3x, + 3x,- -11
2x, - 3x, + x, + 3, - - 2, 0
-X, + 4x, - 4x, - 6, + 2x, + 4x,- -9
3x, - , + Sx, + 2x, - 3x,
- 2r, + 3x, - 4x, - 6x, + x, + 2x, - - 12
and Fis the 3 x 3 flexibility matrix for the beam, to find
the stiffness matrix F- and the force matrix w. The
entries of F are measured in inches per pound.
0.008 0.004 0.003
61. F-0.004 0.006 0.004
0.003 0.004 0.008
[0.017 0.010 0.008
62. F= 0.010 0.012 0.010. d =0.15
0.008 0.010 0.017]
[0.585
d-0.640
0.835
[ 0]
of
63. Proof Prove Property 2 of Theorem 2.8: If A is an
invertible matrix and is a positive integer, then
52.
(A)- - A'A-. A- (A-Y
k factors
5x, - I
64. Proof Prove Property 4 of Theorem 2.8: If A is an
invertible matrix, then (A)-- (A-.
65. Proof Prove Property 2 of Theorem 2.10: If C is an
invertible matrix such that CA - CB, then A - B.
66. Proof Prove that if A= A, then
Matrix Equal to Its Own Inverse In Exercises 53 and
54, find r such that the matrix is equal to its own inverse.
53. A -|
54. A-i
1- 24 = (1 - 24)-1.
C Cm e
y m C l
2.3 Exercises
73
67. Guided Proof Prove that the inverse of a symmetric
nonsingular matrix is symmetric.
Getting Started: To prove that the inverse of A is
symmetric, you need to show that (A-)7 = A-.
(i) Let A be a symmetric, nonsingular matrix.
(ii) This means that A = A and A exists.
75. Use the result of Exercise 74 to find A for each
matrix.
[-1
(a) A-
3
0 0
2
(iii) Use the properties of the transpose to show that
(A- is equal to A.
(b) A =0
68. Proof Prove that if A, B, and C are square matrices
and ABC = 1, then B is invertible and B- CA.
76. Let A=
69. Proof Prove that if A is invertible and AB - 0,
then B = 0.
70. Guided Proof Prove that if A - A, then either A is
singular or A-.
(a) Show that A - 24 + 51 - O, where I is the
identity matrix of order 2.
(b) Show that A-- (21 - A).
(c) Show that for any square matrix satisfying
A - 24 + 5/- 0, the inverse of A is
Getting Started: You must show that either A is singular
or A equals the identity matrix.
(i) Begin your proof by observing that A is either
singular or nonsingular.
A-- (21 - A).
77. Proof Let u be an nxI column matrix satisfying
u'u - . The x matrix H - 1, - 2uu' is called a
Householder matrix.
(ii) If A is singular, then you are done.
(ii) If A is nonsingular, then use the inverse matrix A-
and the hypothesis A A to show thatA-/
(a) Prove that H is symmetric and nonsingular.
True or False? In Exercises 71 and 72, determine
whether each statement is true or false. If a statement
(b) Let u =2/2 Show that u'u = 1, and find the
is true, give a reason or cite an appropriate statement
from the text. If a statement is false, provide an exampl
Houscholder matrix H](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F334d3280-5be3-488a-979d-963cfd5dbab4%2F3fd4ig_processed.jpeg&w=3840&q=75)
Transcribed Image Text:0.6
21. 0.7
-0.3]
0.1
0.2 0.3
22. -0.3
0.2
43. A-
B--
-1
0.2
0.2
-0.9
0.5
0.5
0.
-4
6
44. A -0
B--2
4
23. 3
2
24. 3
2
4
4
3
4
5
5
5 5
Ce tcer e
t dy C
72
Chapter 2 Matrices
Solving a System of Equations Using an Inverse
In Exercises 45-48, use an inverse matrix to solve each
system of linear equations.
Singular Matrix In Exercises 55 and 56, find x such
that the matrix is singular.
.
55. A -[-
56. A -
46. (a) 2x - y- -3
2r +y= 7
45. (a) x+ 2y --
x- 2y - 3
Solving a Matrix Equation In Exercises 57 and 58,
find A.
(b) x+ 2y - 10
(b) 2x - y--I
x- 2y - -6
2x +y--3
57. (24)- -
58. (44)- -
47. (a) x, + 2x, + x,- 2
X + 2x, - - 4
X - 2x, + --2
(b) x, + 2x, + x,- I
X, + 2x, - x,- 3
X- 2x, + x, --3
48. (a) x, + X, - 2x, - 0
X- 2x, + x - 0
X,- X,- x,--1
(b) x, + x - 2x, - -I
,- 2x, + , - 2
Finding the Inverse of a Matrix In Exercises 59
and 60, show that the matrix is invertible and find its
inverse.
59, a - [-
[sec e tan e
sin e co f
|- cos 6
sin e 60. A
sin e
I
tan e sec e
Beam Deflection In Exercises 61 and 62, forces w.W
and w, (in pounds) act on a simply supported elastic
beam, resulting in deflections d, dz, and d, (in inches) in
the beam (see figure).
Solving a System of Equations Using an Inverse
In Exercises 49-52, use a software program or a
graphing utility to solve the system of linear equations
using an inverse matrix.
49. X, + 2x, - , + 3x, - --3
X - 3x, + x, + 2x, - X = -3
2r, + x + 1, - 3x, + x, = 6
X- + 2x, + -X= 2
2x, + -A + 2x, + X- -3
Use the matrix equation d- Fw, where
--E
d =
50. , + x - x, + 3x, - x, - 3
2x, + 1 + + + I = 4
X, + X - x + 2x,- x,- 3
2x, +x, + 4x, + x- x, - -1
3x, +x+ - 2, + x- 5
51. 2r, - 3x, + x - 2, + Xg - 4xg = 20
3x, + - 4x, + - + 2x,-- 16
4x, + - 3x, + 4, - + 2,- -12
-Sx, - + 4x, + 2x, - 5x, + 3x, -2
X+ - 3x,+ 4 - 3x, + X--15
3x, - X + 2x, - 3x, + 2r, - 6r, - 25
4x, - 2x, + 4x, + 2, - Sx, - - I
3x, + 6x, - Sx, - 6, + 3x, + 3x,- -11
2x, - 3x, + x, + 3, - - 2, 0
-X, + 4x, - 4x, - 6, + 2x, + 4x,- -9
3x, - , + Sx, + 2x, - 3x,
- 2r, + 3x, - 4x, - 6x, + x, + 2x, - - 12
and Fis the 3 x 3 flexibility matrix for the beam, to find
the stiffness matrix F- and the force matrix w. The
entries of F are measured in inches per pound.
0.008 0.004 0.003
61. F-0.004 0.006 0.004
0.003 0.004 0.008
[0.017 0.010 0.008
62. F= 0.010 0.012 0.010. d =0.15
0.008 0.010 0.017]
[0.585
d-0.640
0.835
[ 0]
of
63. Proof Prove Property 2 of Theorem 2.8: If A is an
invertible matrix and is a positive integer, then
52.
(A)- - A'A-. A- (A-Y
k factors
5x, - I
64. Proof Prove Property 4 of Theorem 2.8: If A is an
invertible matrix, then (A)-- (A-.
65. Proof Prove Property 2 of Theorem 2.10: If C is an
invertible matrix such that CA - CB, then A - B.
66. Proof Prove that if A= A, then
Matrix Equal to Its Own Inverse In Exercises 53 and
54, find r such that the matrix is equal to its own inverse.
53. A -|
54. A-i
1- 24 = (1 - 24)-1.
C Cm e
y m C l
2.3 Exercises
73
67. Guided Proof Prove that the inverse of a symmetric
nonsingular matrix is symmetric.
Getting Started: To prove that the inverse of A is
symmetric, you need to show that (A-)7 = A-.
(i) Let A be a symmetric, nonsingular matrix.
(ii) This means that A = A and A exists.
75. Use the result of Exercise 74 to find A for each
matrix.
[-1
(a) A-
3
0 0
2
(iii) Use the properties of the transpose to show that
(A- is equal to A.
(b) A =0
68. Proof Prove that if A, B, and C are square matrices
and ABC = 1, then B is invertible and B- CA.
76. Let A=
69. Proof Prove that if A is invertible and AB - 0,
then B = 0.
70. Guided Proof Prove that if A - A, then either A is
singular or A-.
(a) Show that A - 24 + 51 - O, where I is the
identity matrix of order 2.
(b) Show that A-- (21 - A).
(c) Show that for any square matrix satisfying
A - 24 + 5/- 0, the inverse of A is
Getting Started: You must show that either A is singular
or A equals the identity matrix.
(i) Begin your proof by observing that A is either
singular or nonsingular.
A-- (21 - A).
77. Proof Let u be an nxI column matrix satisfying
u'u - . The x matrix H - 1, - 2uu' is called a
Householder matrix.
(ii) If A is singular, then you are done.
(ii) If A is nonsingular, then use the inverse matrix A-
and the hypothesis A A to show thatA-/
(a) Prove that H is symmetric and nonsingular.
True or False? In Exercises 71 and 72, determine
whether each statement is true or false. If a statement
(b) Let u =2/2 Show that u'u = 1, and find the
is true, give a reason or cite an appropriate statement
from the text. If a statement is false, provide an exampl
Houscholder matrix H
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