Singular Matrix In Exercises 55 and 56, find z such that the matrix is singular. 55. A - 56. A --

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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#55
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
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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