#55 0.621. 0.7-0.3]0.10.2 0.322. -0.30.243. A-B---10.20.2-0.90.50.50.-4644. A -0B--2423. 3224. 324434555 5Ce tcer et dy C72Chapter 2 MatricesSolving a System of Equations Using an InverseIn Exercises 45-48, use an inverse matrix to solve eachsystem of linear equations.Singular Matrix In Exercises 55 and 56, find x suchthat the matrix is singular..55. A -[-56. A -46. (a) 2x - y- -32r +y= 745. (a) x+ 2y --x- 2y - 3Solving a Matrix Equation In Exercises 57 and 58,find A.(b) x+ 2y - 10(b) 2x - y--Ix- 2y - -62x +y--357. (24)- -58. (44)- -47. (a) x, + 2x, + x,- 2X + 2x, - - 4X - 2x, + --2(b) x, + 2x, + x,- IX, + 2x, - x,- 3X- 2x, + x, --348. (a) x, + X, - 2x, - 0X- 2x, + x - 0X,- X,- x,--1(b) x, + x - 2x, - -I,- 2x, + , - 2Finding the Inverse of a Matrix In Exercises 59and 60, show that the matrix is invertible and find itsinverse.59, a - [-[sec e tan esin e co f|- cos 6sin e 60. Asin eItan e sec eBeam Deflection In Exercises 61 and 62, forces w.Wand w, (in pounds) act on a simply supported elasticbeam, resulting in deflections d, dz, and d, (in inches) inthe beam (see figure).Solving a System of Equations Using an InverseIn Exercises 49-52, use a software program or agraphing utility to solve the system of linear equationsusing an inverse matrix.49. X, + 2x, - , + 3x, - --3X - 3x, + x, + 2x, - X = -32r, + x + 1, - 3x, + x, = 6X- + 2x, + -X= 22x, + -A + 2x, + X- -3Use the matrix equation d- Fw, where--Ed =50. , + x - x, + 3x, - x, - 32x, + 1 + + + I = 4X, + X - x + 2x,- x,- 32x, +x, + 4x, + x- x, - -13x, +x+ - 2, + x- 551. 2r, - 3x, + x - 2, + Xg - 4xg = 203x, + - 4x, + - + 2x,-- 164x, + - 3x, + 4, - + 2,- -12-Sx, - + 4x, + 2x, - 5x, + 3x, -2X+ - 3x,+ 4 - 3x, + X--153x, - X + 2x, - 3x, + 2r, - 6r, - 254x, - 2x, + 4x, + 2, - Sx, - - I3x, + 6x, - Sx, - 6, + 3x, + 3x,- -112x, - 3x, + x, + 3, - - 2, 0-X, + 4x, - 4x, - 6, + 2x, + 4x,- -93x, - , + Sx, + 2x, - 3x,- 2r, + 3x, - 4x, - 6x, + x, + 2x, - - 12and Fis the 3 x 3 flexibility matrix for the beam, to findthe stiffness matrix F- and the force matrix w. Theentries of F are measured in inches per pound.0.008 0.004 0.00361. F-0.004 0.006 0.0040.003 0.004 0.008[0.017 0.010 0.00862. F= 0.010 0.012 0.010. d =0.150.008 0.010 0.017][0.585d-0.6400.835[ 0]of63. Proof Prove Property 2 of Theorem 2.8: If A is aninvertible matrix and is a positive integer, then52.(A)- - A'A-. A- (A-Yk factors5x, - I64. Proof Prove Property 4 of Theorem 2.8: If A is aninvertible matrix, then (A)-- (A-.65. Proof Prove Property 2 of Theorem 2.10: If C is aninvertible matrix such that CA - CB, then A - B.66. Proof Prove that if A= A, thenMatrix Equal to Its Own Inverse In Exercises 53 and54, find r such that the matrix is equal to its own inverse.53. A -|54. A-i1- 24 = (1 - 24)-1.C Cm ey m C l2.3 Exercises7367. Guided Proof Prove that the inverse of a symmetricnonsingular matrix is symmetric.Getting Started: To prove that the inverse of A issymmetric, 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 eachmatrix.[-1(a) A-30 02(iii) Use the properties of the transpose to show that(A- is equal to A.(b) A =068. Proof Prove that if A, B, and C are square matricesand 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 issingular or A-.(a) Show that A - 24 + 51 - O, where I is theidentity matrix of order 2.(b) Show that A-- (21 - A).(c) Show that for any square matrix satisfyingA - 24 + 5/- 0, the inverse of A isGetting Started: You must show that either A is singularor A equals the identity matrix.(i) Begin your proof by observing that A is eithersingular or nonsingular.A-- (21 - A).77. Proof Let u be an nxI column matrix satisfyingu'u - . The x matrix H - 1, - 2uu' is called aHouseholder 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, determinewhether each statement is true or false. If a statement(b) Let u =2/2 Show that u'u = 1, and find theis true, give a reason or cite an appropriate statementfrom the text. If a statement is false, provide an examplHouscholder matrix H

Name: #55 0.621. 0.7-0.3]0.10.2 0.322. -0.30.243. A-B---10.20.2-0.90.50.50.-
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Description: #55 0.621. 0.7-0.3]0.10.2 0.322. -0.30.243. A-B---10.20.2-0.90.50.50.-4644. A -0B--2423. 3224. 324434555 5Ce tcer et dy C72Chapter 2 MatricesSolving a System of Equations Using an InverseIn Exercises 45-48, use an inverse matrix to solve eachsystem o

For (55), We are given the matrix A=4x-2-3. Here, we need to find x such that the matrix is…

Answered: Singular Matrix In Exercises 55 and 56,…

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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

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-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
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2
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4
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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

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