#65 12:54Done elementary_linear_algebra_8th_edi...number of cach denomination.A System of Linear Equations In Exercises 39-42, useOa software program or a graphing utility to solve thesystem of linear equations.Matrix Representation In Exercises 49 and 50, assumethat the matrix is the augmented matrix of a systemof linear equations, and (a) determine the number ofequations and the number of variables, and (b) find thevalue(s) of k such that the system is consistent. Thenassume that the matrix is the coefficient matrix of ahomogeneous system of linear equations, and repeatparts (a) and (b).39. X, - 2x, + 5x, - 3x, - 23.6X, + 4x, - 7x, - 2x, - 45.73x, - Sx, + 7x, + 4,- 29.940. X, + - 2x, + 3, + 2x, -93x, + 3x, - x, + + , 52x, + 2x, - x, + x- 2r, - 14x, + 4, + x,&r, + Sx, - 2x, - ,+ 2x, - 341. x, - x, + 2x, + 2r, + 6r,- 63x, - 2x, + 4x, + 4x, + 12, 14X- - - 3,- -32x, - 2x, + 4x, + Sx, + 15x, - 102r, - 2x, + 4x,+ 4x, + 13x, - 1342. X, + 2x, - 2x, + 2x, - X, + 3,- 02x,- x + 3x, + - 3x, + 2x, - 17X + 3x, - 21, + - 2x, - 3x,- -53x, - 2x, + x, - x + 3x, - 2x, --1-X, - 2x, + X+ 2x, - 2x, + 3- 10X, - 3x, + , + 3x, - 2x, + -II249. A-- 3,-4-322 -1 350. A--44 -2Coefficient Design In Exercises 51 and 52, find valuesof a, b, and e (if possible) such that the system of linearequations has (a) a unique solution, (b) no solution, and(e) infinitely many solutions.51. x+ y-252. + v-0y+ -2* + :- 2ar + by + cz -0y+ -0* +-0ar + by + cz = oCe tcel ey C24Chapter 1 Systems of Linear Equations53. The system below has one solution: x= 1. y= -1,and : - 2.61. Writing Is it possible for a system of linear equationswith fewer equations than variables to have no solution?If so, give an example.62. Writing Does a matrix have a unique row-echelonform? Illustrate your answer with examples.4x - 2y + 5z = 16Equation IEquation 2-x- 3y + 2:- 6 Bquation 3I+ y -0Row Equivalence In Exercises 63 and 64, determineSolve the systems provided by (a) Equations I and 2,(b) Equations I and 3, and (c) Equations 2 and 3.(d) How many solutions does each of these systemshave?conditions on a, b, c, and d such that the matrix54. Assume the system below has a unique solution.will be row-equivalent to the given matrix.a + a, + a,-b,a + a + at- b;ak, + a + al = b,Equation IEquation 2Equation 363. : |64.Homogeneous System In Exercises 65 and 66, findall values of A (the Greek letter lambda) for which theDoes the system composed of Equations I and 2 havea unique solution, no solution, or infinitely manysolutions?homogeneous linear system has nontrivial solutions.65. A - 2)x +y-0Row Equivalence In Exercises 55 and 56, find thereduced row-echelon matrix that is row-equivalent tothe given matrix.x+ (A - 2y - 066. (21 + 9)x - Sy -0I- Ay-02 3]ss.67. The augmented matrix represents a system of lincarequations that has been reduced using Gauss-Jordanelimination. Write a system of equations with nonzerocoefficients that the reduced matrix could represent.56. 459857. Writing Describe all possible 2x2 reducedrow-echelon matrices. Support your answer withexamples.58. Writing Describe all possible 3 x 3 reducedrow-echelon matrices. Support your answer withexamples.3 -24There are many correct answers.True or False? In Exercises 59 and 60, determine68. CAPSTONE In your own words, describe thedifference between a matrix in row-echelon formand a matrix in reduced row-echelon form. Includewhether each statement is true or false. If a statementis true, give a reason or cite an appropriate statementfrom the text. If a statement is false, provide an examplean example of cach to support your explanation.that shows the statement is not true in all cases or cite anappropriate statement from the text.59. (a) A6x 3 matrix has six rows.69. Writing Consider the ? x ? matrix :(b) Every matrix is row-equivalent to a matrix inrow-echelon form.Perform the sequence of row operations.(a) Add (-1) times the second row to the first row.(e) If the row-echelon form of the augmented matrixof a system of lincar equations contains the row(10000l, then the original system is inconsistent.(b) Add I times the first row to the second row(c) Add (-1) times the second row to the first row.(d) A homogeneous system of four lincar equations insix variables has infinitely many solutions.(d) Multiply the first row by (-1).What happened to the original matrix? Describe, ingeneral, how to interchange two rows of a matrix usingonly the second and third elementary row operations.60. (a) A 4 x 7 matrix has four columns.(b) Every matrix has a unique reduced row-echelon form.

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Description: #65 12:54Done elementary_linear_algebra_8th_edi...number of cach denomination.A System of Linear Equations In Exercises 39-42, useOa software program or a graphing utility to solve thesystem of linear equations.Matrix Representation In Exercises 49 a

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of a, b, and e (if possible) such that the system of linear equations has (a) a unique solution, (b) no solution, and (e) infinitely many solutions. -2 y+ -2 17 S1. + y 52. + y -0 -1 y+ -0 10 ax + by + ez- 0 ar + by + ez -0 tions 61. Writing Is it possible for a system of linear equations with fewer equations than variables to have no solution? If so give an esample

of a, b, and e (if possible) such that the system of linear equations has (a) a unique solution, (b) no solution, and (e) infinitely many solutions. -2 y+ -2 17 S1. + y 52. + y -0 -1 y+ -0 10 ax + by + ez- 0 ar + by + ez -0 tions 61. Writing Is it possible for a system of linear equations with fewer equations than variables to have no solution? If so give an esample

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Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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#65

12:54
Done elementary_linear_algebra_8th_edi...
number of cach denomination.
A System of Linear Equations In Exercises 39-42, use
Oa software program or a graphing utility to solve the
system of linear equations.
Matrix Representation In Exercises 49 and 50, assume
that the matrix is the augmented matrix of a system
of linear equations, and (a) determine the number of
equations and the number of variables, and (b) find the
value(s) of k such that the system is consistent. Then
assume that the matrix is the coefficient matrix of a
homogeneous system of linear equations, and repeat
parts (a) and (b).
39. X, - 2x, + 5x, - 3x, - 23.6
X, + 4x, - 7x, - 2x, - 45.7
3x, - Sx, + 7x, + 4,- 29.9
40. X, + - 2x, + 3, + 2x, -9
3x, + 3x, - x, + + , 5
2x, + 2x, - x, + x- 2r, - 1
4x, + 4, + x,
&r, + Sx, - 2x, - ,+ 2x, - 3
41. x, - x, + 2x, + 2r, + 6r,- 6
3x, - 2x, + 4x, + 4x, + 12, 14
X- - - 3,- -3
2x, - 2x, + 4x, + Sx, + 15x, - 10
2r, - 2x, + 4x,+ 4x, + 13x, - 13
42. X, + 2x, - 2x, + 2x, - X, + 3,- 0
2x,- x + 3x, + - 3x, + 2x, - 17
X + 3x, - 21, + - 2x, - 3x,- -5
3x, - 2x, + x, - x + 3x, - 2x, --1
-X, - 2x, + X+ 2x, - 2x, + 3- 10
X, - 3x, + , + 3x, - 2x, + -II
2
49. A-
- 3,-4
-3
2
2 -1 3
50. A--4
4 -2
Coefficient Design In Exercises 51 and 52, find values
of a, b, and e (if possible) such that the system of linear
equations has (a) a unique solution, (b) no solution, and
(e) infinitely many solutions.
51. x+ y
-2
52. + v
-0
y+ -2
* + :- 2
ar + by + cz -0
y+ -0
* +-0
ar + by + cz = o
Ce tcel e
y C
24
Chapter 1 Systems of Linear Equations
53. The system below has one solution: x= 1. y= -1,
and : - 2.
61. Writing Is it possible for a system of linear equations
with fewer equations than variables to have no solution?
If so, give an example.
62. Writing Does a matrix have a unique row-echelon
form? Illustrate your answer with examples.
4x - 2y + 5z = 16
Equation I
Equation 2
-x- 3y + 2:- 6 Bquation 3
I+ y -0
Row Equivalence In Exercises 63 and 64, determine
Solve the systems provided by (a) Equations I and 2,
(b) Equations I and 3, and (c) Equations 2 and 3.
(d) How many solutions does each of these systems
have?
conditions on a, b, c, and d such that the matrix
54. Assume the system below has a unique solution.
will be row-equivalent to the given matrix.
a + a, + a,-b,
a + a + at- b;
ak, + a + al = b,
Equation I
Equation 2
Equation 3
63. : |
64.
Homogeneous System In Exercises 65 and 66, find
all values of A (the Greek letter lambda) for which the
Does the system composed of Equations I and 2 have
a unique solution, no solution, or infinitely many
solutions?
homogeneous linear system has nontrivial solutions.
65. A - 2)x +
y-0
Row Equivalence In Exercises 55 and 56, find the
reduced row-echelon matrix that is row-equivalent to
the given matrix.
x+ (A - 2y - 0
66. (21 + 9)x - Sy -0
I- Ay-0
2 3]
ss.
67. The augmented matrix represents a system of lincar
equations that has been reduced using Gauss-Jordan
elimination. Write a system of equations with nonzero
coefficients that the reduced matrix could represent.
56. 4
5
9
8
57. Writing Describe all possible 2x2 reduced
row-echelon matrices. Support your answer with
examples.
58. Writing Describe all possible 3 x 3 reduced
row-echelon matrices. Support your answer with
examples.
3 -2
4
There are many correct answers.
True or False? In Exercises 59 and 60, determine
68. CAPSTONE In your own words, describe the
difference between a matrix in row-echelon form
and a matrix in reduced row-echelon form. Include
whether each statement is true or false. If a statement
is true, give a reason or cite an appropriate statement
from the text. If a statement is false, provide an example
an example of cach to support your explanation.
that shows the statement is not true in all cases or cite an
appropriate statement from the text.
59. (a) A6x 3 matrix has six rows.
69. Writing Consider the ? x ? matrix :
(b) Every matrix is row-equivalent to a matrix in
row-echelon form.
Perform the sequence of row operations.
(a) Add (-1) times the second row to the first row.
(e) If the row-echelon form of the augmented matrix
of a system of lincar equations contains the row
(10000l, then the original system is inconsistent.
(b) Add I times the first row to the second row
(c) Add (-1) times the second row to the first row.
(d) A homogeneous system of four lincar equations in
six variables has infinitely many solutions.
(d) Multiply the first row by (-1).
What happened to the original matrix? Describe, in
general, how to interchange two rows of a matrix using
only the second and third elementary row operations.
60. (a) A 4 x 7 matrix has four columns.
(b) Every matrix has a unique reduced row-echelon form.

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