36. nd a Verify that det AB = (det A) (det B) for the matrices in Exercises 37 and 38. (Do not use Theorem 6.) HI 3 = [³ 6 37. A = 0 2 i] B = [3 5 0 4]

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
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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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37

26.
21 2
-6
0000
3
7
3
[]
In Exercises 27 and 28, A and B are n x n matrices. Mark each
statement True or False. Justify each answer.
5
d.
28 a.
-6
4
27. a. A row replacement operation does not affect the determi-
nant of a matrix.
b. The determinant of A is the product of the pivots in any
echelon form U of A, multiplied by (-1)", where r is the
number of row interchanges made during row reduction
from A to U.
c. If the columns of A are linearly dependent, then
det A = 0.
det(A + B) = det A + det B
If three row interchanges are made in succession, then the
new determinant equals the old determinant.
-2
b.
The determinant of A is the product of the diagonal entries
in A.
c. If det A is zero, then two rows or two columns are the
same, or a row or a column is zero.
d. det A¹ = (-1) det A.
29. Compute det B4, where B =
1
-1
1
1
30. Use Theorem 3 (but not Theorem 4) to show that if two rows
of a square matrix A are equal, then det A = 0. The same is
true for two columns. Why?
37. A =
0
1
1
2
2 1
In Exercises 31-36, mention an appropriate theorem in your
explanation.
1
det A
32. Suppose that A is a square matrix such that det A³ = 0.
Explain why A cannot be invertible.
31, Show that if A is invertible, then det A-¹ =
-1 -2]'
L-
[-1 -3]
39. Let A and B be 3 x 3 matrices, with det F
det B 4. Use properties of determinants (in
in the exercises above) to compute:
a. det AB
b. det 5A
d.
det A-1
e. det A³
det B = -1. Compute:
40, Let A and B be 4 x 4 matrices, with de
b. det B³ c. det
a. det AB
B5
det AT BA
-1
e. det B-¹AB
d.
41. Verify that det A = det B + det C, where
a +e
C
d
C
f
[ª
b + 1]. B = [a b]
- [a a b
1
[]
and B =
1
d
det(A + B) = det A + det B if and only if
43. Verify that det A = det B + det C, where
33. Let A and B be square matrices. Show that even though
AB and BA may not be equal, it is always true that
det AB = det BA.
2
[ 3 i]· B = [ ³²
6
34. Let A and P be square matrices, with P invertible. Show that
det(PAP-¹) = det A.
35. Let U be a square matrix such that UTU = I. Show that
det U = ±1.
36. Find a formula for det(rA) when A is an n x n matrix.
Verify that det AB = (det A) (det B) for the matrices in Exercises
37 and 38. (Do not use Theorem 6.)
H
4
PIR
38. A =
A =
=
42. Let A =
A =
a11
a21
a31
B =
a 12
a22
a 32
U₁ + v₁
U₂ + V₂
น2
U3 + V3
a12
UI
a11
a22
U2
a21
a31
a32
Uz
A31
Note, however, that A is not the same as
c. det E
C =
a11
A =
a21
44. Right-multiplication by an elementary m
columns of A in the same way that left-m
the rows. Use Theorems 5 and 3 and the c
is another elementary matrix to show tha
det AE = (det E) (det A
Do not use Theorem 6.
45. [M] Compute det AT A and det AAT
4 x 5 matrices and several random 5 x
you say about AT A and AAT when A ha
rows?
46. [M] If det A is close to zero, is the mat
Experiment with the nearly singular 4
4 0 -7
-6 1 11
7 -5 10
-1 2 3
Compute the determinants of A, 10A
compute the condition numbers of
these calculations when A is the 4x
cuss your results.
Transcribed Image Text:26. 21 2 -6 0000 3 7 3 [] In Exercises 27 and 28, A and B are n x n matrices. Mark each statement True or False. Justify each answer. 5 d. 28 a. -6 4 27. a. A row replacement operation does not affect the determi- nant of a matrix. b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)", where r is the number of row interchanges made during row reduction from A to U. c. If the columns of A are linearly dependent, then det A = 0. det(A + B) = det A + det B If three row interchanges are made in succession, then the new determinant equals the old determinant. -2 b. The determinant of A is the product of the diagonal entries in A. c. If det A is zero, then two rows or two columns are the same, or a row or a column is zero. d. det A¹ = (-1) det A. 29. Compute det B4, where B = 1 -1 1 1 30. Use Theorem 3 (but not Theorem 4) to show that if two rows of a square matrix A are equal, then det A = 0. The same is true for two columns. Why? 37. A = 0 1 1 2 2 1 In Exercises 31-36, mention an appropriate theorem in your explanation. 1 det A 32. Suppose that A is a square matrix such that det A³ = 0. Explain why A cannot be invertible. 31, Show that if A is invertible, then det A-¹ = -1 -2]' L- [-1 -3] 39. Let A and B be 3 x 3 matrices, with det F det B 4. Use properties of determinants (in in the exercises above) to compute: a. det AB b. det 5A d. det A-1 e. det A³ det B = -1. Compute: 40, Let A and B be 4 x 4 matrices, with de b. det B³ c. det a. det AB B5 det AT BA -1 e. det B-¹AB d. 41. Verify that det A = det B + det C, where a +e C d C f [ª b + 1]. B = [a b] - [a a b 1 [] and B = 1 d det(A + B) = det A + det B if and only if 43. Verify that det A = det B + det C, where 33. Let A and B be square matrices. Show that even though AB and BA may not be equal, it is always true that det AB = det BA. 2 [ 3 i]· B = [ ³² 6 34. Let A and P be square matrices, with P invertible. Show that det(PAP-¹) = det A. 35. Let U be a square matrix such that UTU = I. Show that det U = ±1. 36. Find a formula for det(rA) when A is an n x n matrix. Verify that det AB = (det A) (det B) for the matrices in Exercises 37 and 38. (Do not use Theorem 6.) H 4 PIR 38. A = A = = 42. Let A = A = a11 a21 a31 B = a 12 a22 a 32 U₁ + v₁ U₂ + V₂ น2 U3 + V3 a12 UI a11 a22 U2 a21 a31 a32 Uz A31 Note, however, that A is not the same as c. det E C = a11 A = a21 44. Right-multiplication by an elementary m columns of A in the same way that left-m the rows. Use Theorems 5 and 3 and the c is another elementary matrix to show tha det AE = (det E) (det A Do not use Theorem 6. 45. [M] Compute det AT A and det AAT 4 x 5 matrices and several random 5 x you say about AT A and AAT when A ha rows? 46. [M] If det A is close to zero, is the mat Experiment with the nearly singular 4 4 0 -7 -6 1 11 7 -5 10 -1 2 3 Compute the determinants of A, 10A compute the condition numbers of these calculations when A is the 4x cuss your results.
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