Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
Related questions
Question
12,14,16,30,50 please show work
![1.2
EXERCISES
Consider the matrices in Exercises 1-10.
a) Either state that the matrix is in echelon form
or use elementary row operations to
transform it to echelon form.
b) If the matrix is in echelon form, transform it
to reduced echelon form.
1. 12
2.1 2-1
0 1 3
[83]
*[388]
41
0023
2014
3.
[
5.
7.
9.
10.
1
1
w
0142
001 1
21
[
7
77
02-2-3
0 0 0 1
-1 4-3 4 6
01
*[83]
2
021-3-3
00012
8.
31
0012
2-1
0
0
0-3
In Exercises 11-21, each of the given matrices represents
the augmented matrix for a system of linear equations.
In each exercise, display the solution set or state that the
system is inconsistent.
11. 1 10
0 10
13.
15.
17.
1 210
0131
1110
0100
0001
19.100 1
20.
0101
0001
112020
011100
001212
21. 213201
001121
000030
10100 18.
0011
00010
12.
22. 2x₁3x₂ = 5
-4x₁ + 6x₂ = -10
14.
23. x₁2x₂ = 3
2x₁ - 4x₂ = 1
16.
002
1221
01 00
1201
01 10
In Exercises 22-35, solve the system by transforming
the augmented matrix to reduced echelon form.
0020
1213
0002
0000](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc0db768-dd44-4b93-bf17-25d90f2dc46a%2F9c137df2-c420-4ef4-a5af-fc68638a604e%2Foaz3ccp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.2
EXERCISES
Consider the matrices in Exercises 1-10.
a) Either state that the matrix is in echelon form
or use elementary row operations to
transform it to echelon form.
b) If the matrix is in echelon form, transform it
to reduced echelon form.
1. 12
2.1 2-1
0 1 3
[83]
*[388]
41
0023
2014
3.
[
5.
7.
9.
10.
1
1
w
0142
001 1
21
[
7
77
02-2-3
0 0 0 1
-1 4-3 4 6
01
*[83]
2
021-3-3
00012
8.
31
0012
2-1
0
0
0-3
In Exercises 11-21, each of the given matrices represents
the augmented matrix for a system of linear equations.
In each exercise, display the solution set or state that the
system is inconsistent.
11. 1 10
0 10
13.
15.
17.
1 210
0131
1110
0100
0001
19.100 1
20.
0101
0001
112020
011100
001212
21. 213201
001121
000030
10100 18.
0011
00010
12.
22. 2x₁3x₂ = 5
-4x₁ + 6x₂ = -10
14.
23. x₁2x₂ = 3
2x₁ - 4x₂ = 1
16.
002
1221
01 00
1201
01 10
In Exercises 22-35, solve the system by transforming
the augmented matrix to reduced echelon form.
0020
1213
0002
0000
![24. X₁ X₂ + x3 = 3
2x₁ + x₂ - 4x3 = -3
25. x₁ + x₂ = 2
3x₁ + 3x₂ = 6
26. X₁
X₂ + x3 = 4
2x₁2x2 + 3x3 = 2
27. X₁ + X₂
X3 = 2
-3x₁-3x2 + 3x3 = -6
28. 2x₁ +
3x₂ - 4x3 = 3
x₁ =
2x₂ - 2x3 = -2
-x1 + 16x2 + 2x3 = 16
29. X₁ + X₂ X3 =
1
2x₁ - x₂ + 7x3 =
8
-x₁ + x₂ = 5x3 = -5
30. x₁ + x₂
XI
31. x₁
- X5 = 1
X₂ + 2x3 + x4 + 3x5 = 1
- x3 + x4 + xs = 0
+ x3 + x4-2x5=1
2x₁ + x₂ + 3x3 x4 + X5 = 0
3x₁ - x₂ + 4x3 + x4 +
X5 = 1
32. x₁ + x₂ = 1
x₁ - x₂ = 3
2x₁ + x₂ = 3
34. x₁ + 2x₂ = 1
2x₁ + 4x₂ = 2
-X₁ - 2x₂ = -1
36. x1 + 2x2 = -3
axy - 2x₂ = 5
38. 2x₁ + 4x₂ = a
3x₁ + 6x₂ = 5
40. x₁ + ax₂ = 6
ax₁ + 2ax₂ = 4
33.
41. 2 cosa + 4 sin ß = 3
3 cos a -5 sin ß = -1
42. 2 cos a
x₁ + x₂ = 1
x₁ - x₂ = 3
2x₁ + x₂ = 2
X₂
35. X₁
x₁
In Exercises 36-40, find all values a for which the sys-
tem has no solution.
37. X1 + 3x2 = 4
2x₁ + 6x₂ = a
39. 3x₁ + ax₂ = 3
ax₁ + 3x₂ = 5
sin² ß = 1
12 cos² a + 8 sin² ß = 13
In Exercises 41 and 42, find all values a and 8 where
0 ≤ ≤ 27 and 0 ≤ B ≤ 2.
1.2 Echelon Form and Gauss-Jordan Elimination
X3 = 1
+ x3 = 2
x₂ + 2x3 = 3
For X₁, X₂, X3 in the solution set:
43. Describe the solution set of the following system in
terms of X3:
x₁ + x₂ + x3 = 3
x₁ + 2x2 = 5.
a) Find the maximum value of x3 such that
x₁ ≥ 0 and x₂ > 0.
b) Find the maximum value of
y = 2x₁ - 4x₂ + x3 subject to x₂ > 0 and
X₂ ≥ 0.
c) Find the minimum value of
y = (x₁ - 1)² + (x₂ + 3)² + (x3 + 1)² with no
restriction on x₂ or x2. [Hint: Regard y as a
function of x3 and set the derivative equal to 0;
then apply the second-derivative test to verify
that you have found a minimum.]
44. Let A and I be as follows:
27
A =
= [X %] -=[¦ ¦]·
c b
Prove that if b-cd #0, then A is row equivalent
to I.
45. As in Fig. 1.4, display all the possible configurations
for a (2 x 3) matrix that is in echelon form. [Hint:
There are seven such configurations. Consider the
various positions that can be occupied by one, two,
or none of the symbols.]
46. Repeat Exercise 45 for a (3 x 2) matrix, for a (3 x 3)
matrix, and for a (3 x 4) matrix.
47. Consider the matrices B and C:
-[23]. -=[¦?].
By Exercise 44, B and C are both row equivalent to
matrix / in Exercise 44. Determine elementary row
operations that demonstrate that B is row equivalent
to C.
48. Repeat Exercise 47 for the matrices
B =
*-[BH] <-[H]
=
49. A certain three-digit number N equals fifteen times
the sum of its digits. If its digits are reversed, the
resulting number exceeds N by 396. The one's digit
is one larger than the sum of the other two. Give
a linear system of three equations whose three un-
knowns are the digits of N. Solve the system and
find N.
50. Find the equation of the parabola, y = ax²+bx+c₂
that passes through the points (-1, 6), (1,4), and
(2,9). [Hint: For each point, give a linear equation
in a, b, and c.]
51. Three people play a game in which there are al-
ways two winners and one loser. They have the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc0db768-dd44-4b93-bf17-25d90f2dc46a%2F9c137df2-c420-4ef4-a5af-fc68638a604e%2Fx2k4u3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:24. X₁ X₂ + x3 = 3
2x₁ + x₂ - 4x3 = -3
25. x₁ + x₂ = 2
3x₁ + 3x₂ = 6
26. X₁
X₂ + x3 = 4
2x₁2x2 + 3x3 = 2
27. X₁ + X₂
X3 = 2
-3x₁-3x2 + 3x3 = -6
28. 2x₁ +
3x₂ - 4x3 = 3
x₁ =
2x₂ - 2x3 = -2
-x1 + 16x2 + 2x3 = 16
29. X₁ + X₂ X3 =
1
2x₁ - x₂ + 7x3 =
8
-x₁ + x₂ = 5x3 = -5
30. x₁ + x₂
XI
31. x₁
- X5 = 1
X₂ + 2x3 + x4 + 3x5 = 1
- x3 + x4 + xs = 0
+ x3 + x4-2x5=1
2x₁ + x₂ + 3x3 x4 + X5 = 0
3x₁ - x₂ + 4x3 + x4 +
X5 = 1
32. x₁ + x₂ = 1
x₁ - x₂ = 3
2x₁ + x₂ = 3
34. x₁ + 2x₂ = 1
2x₁ + 4x₂ = 2
-X₁ - 2x₂ = -1
36. x1 + 2x2 = -3
axy - 2x₂ = 5
38. 2x₁ + 4x₂ = a
3x₁ + 6x₂ = 5
40. x₁ + ax₂ = 6
ax₁ + 2ax₂ = 4
33.
41. 2 cosa + 4 sin ß = 3
3 cos a -5 sin ß = -1
42. 2 cos a
x₁ + x₂ = 1
x₁ - x₂ = 3
2x₁ + x₂ = 2
X₂
35. X₁
x₁
In Exercises 36-40, find all values a for which the sys-
tem has no solution.
37. X1 + 3x2 = 4
2x₁ + 6x₂ = a
39. 3x₁ + ax₂ = 3
ax₁ + 3x₂ = 5
sin² ß = 1
12 cos² a + 8 sin² ß = 13
In Exercises 41 and 42, find all values a and 8 where
0 ≤ ≤ 27 and 0 ≤ B ≤ 2.
1.2 Echelon Form and Gauss-Jordan Elimination
X3 = 1
+ x3 = 2
x₂ + 2x3 = 3
For X₁, X₂, X3 in the solution set:
43. Describe the solution set of the following system in
terms of X3:
x₁ + x₂ + x3 = 3
x₁ + 2x2 = 5.
a) Find the maximum value of x3 such that
x₁ ≥ 0 and x₂ > 0.
b) Find the maximum value of
y = 2x₁ - 4x₂ + x3 subject to x₂ > 0 and
X₂ ≥ 0.
c) Find the minimum value of
y = (x₁ - 1)² + (x₂ + 3)² + (x3 + 1)² with no
restriction on x₂ or x2. [Hint: Regard y as a
function of x3 and set the derivative equal to 0;
then apply the second-derivative test to verify
that you have found a minimum.]
44. Let A and I be as follows:
27
A =
= [X %] -=[¦ ¦]·
c b
Prove that if b-cd #0, then A is row equivalent
to I.
45. As in Fig. 1.4, display all the possible configurations
for a (2 x 3) matrix that is in echelon form. [Hint:
There are seven such configurations. Consider the
various positions that can be occupied by one, two,
or none of the symbols.]
46. Repeat Exercise 45 for a (3 x 2) matrix, for a (3 x 3)
matrix, and for a (3 x 4) matrix.
47. Consider the matrices B and C:
-[23]. -=[¦?].
By Exercise 44, B and C are both row equivalent to
matrix / in Exercise 44. Determine elementary row
operations that demonstrate that B is row equivalent
to C.
48. Repeat Exercise 47 for the matrices
B =
*-[BH] <-[H]
=
49. A certain three-digit number N equals fifteen times
the sum of its digits. If its digits are reversed, the
resulting number exceeds N by 396. The one's digit
is one larger than the sum of the other two. Give
a linear system of three equations whose three un-
knowns are the digits of N. Solve the system and
find N.
50. Find the equation of the parabola, y = ax²+bx+c₂
that passes through the points (-1, 6), (1,4), and
(2,9). [Hint: For each point, give a linear equation
in a, b, and c.]
51. Three people play a game in which there are al-
ways two winners and one loser. They have the
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