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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
33
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe5db299-428d-40d7-abe3-e166485c95d7%2F8232b5f7-0bc1-4616-a3f0-60e1769a9936%2Flnq30vo_processed.jpeg&w=3840&q=75)
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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