A given by 1. M₁1 4. M41 12. A = 14. A = 16. A = A = 2-1 3 1 13-1 4 6 24 1 220-2 2. M₂1 5. M34 -1 12 4 23 7 4 2 10 1 2 1 0 3 2 1 200 310 24 2 1 3. M31 6. M43 2-3 2 -⠀ 13. A = 3 1 -1 15. A = 17. A = 1 -1-2 200 132 214 12 15 0300 0412 0314 In Exercises 8-19, calculate the determinant of the given matrix. Use Theorem 3 to state whether the matrix is singular or nonsingular. 8. A = 10. A = -1 23 4 6 2 9. A = 11. A = [ 1 -2 2 1 2 1 4.2 Determinants and the Eigenvalue Problem 289 23. Let A = (a;;) be the (n x n) matrix specified thus: aij = d for i = j and a¡j = 1 for i ‡ j. For n = 2, 3, and 4, show that det(A) = (d 1)"-¹(d - 1 + n). 24. Let A and B be (n x n) matrices. Use Theorems 2 and 3 to give a quick proof of each of the following. a) If either A or B is singular, then AB is singular. b) If AB is singular, then either A or B is singular. 25. Suppose that A is an (n x n) nonsingular matrix, and recall that det(I) = 1, where I is the (n x n) identity matrix. Show that det (A-¹) = 1/det(A). 26. If A and B are (n xn) matrices, then usually AB # BA Nonetheless, argue that always det (AB) =
A given by 1. M₁1 4. M41 12. A = 14. A = 16. A = A = 2-1 3 1 13-1 4 6 24 1 220-2 2. M₂1 5. M34 -1 12 4 23 7 4 2 10 1 2 1 0 3 2 1 200 310 24 2 1 3. M31 6. M43 2-3 2 -⠀ 13. A = 3 1 -1 15. A = 17. A = 1 -1-2 200 132 214 12 15 0300 0412 0314 In Exercises 8-19, calculate the determinant of the given matrix. Use Theorem 3 to state whether the matrix is singular or nonsingular. 8. A = 10. A = -1 23 4 6 2 9. A = 11. A = [ 1 -2 2 1 2 1 4.2 Determinants and the Eigenvalue Problem 289 23. Let A = (a;;) be the (n x n) matrix specified thus: aij = d for i = j and a¡j = 1 for i ‡ j. For n = 2, 3, and 4, show that det(A) = (d 1)"-¹(d - 1 + n). 24. Let A and B be (n x n) matrices. Use Theorems 2 and 3 to give a quick proof of each of the following. a) If either A or B is singular, then AB is singular. b) If AB is singular, then either A or B is singular. 25. Suppose that A is an (n x n) nonsingular matrix, and recall that det(I) = 1, where I is the (n x n) identity matrix. Show that det (A-¹) = 1/det(A). 26. If A and B are (n xn) matrices, then usually AB # BA Nonetheless, argue that always det (AB) =
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
![A given by
1. M₁1
4. M41
12. A =
14. A =
16. A =
A =
2-1 3 1
4 13-1
624 1
2 20-2
2. M₂1
5. M34
12 4
23 7
4 2 10
12 1
032
-1 1 1
200
310
24 2
3. M31
6. M43
13. A =
15. A =
17. A =
(7)
2-3 2
1
-1 -2
3
200
132
214
1215
0300
0412
0314
In Exercises 8-19, calculate the determinant of the given
matrix. Use Theorem 3 to state whether the matrix is
singular or nonsingular.
31]
-1 2
2 3
46
8. A =
10. A =
9. A =
11. A =
1 -1
-2
21
2
1
1
21
4.2 Determinants and the Eigenvalue Problem
289
23. Let A = (aij) be the (n × n) matrix specified thus:
aij = d for i = j and a¡j = 1 for i # j. For n = 2,
3, and 4, show that
det(A) = (d 1)"-¹ (d - 1 + n).
24. Let A and B be (n x n) matrices. Use Theorems 2
and 3 to give a quick proof of each of the following.
a) If either A or B is singular, then AB is singular.
b) If AB is singular, then either A or B is singular.
25. Suppose that A is an (n × n) nonsingular matrix,
and recall that det (I) = 1, where I is the (n x n)
identity matrix. Show that det(A-¹) = 1/det(A).
26. If A and B are (n × n) matrices, then usually AB ‡
BA. Nonetheless, argue that always det(AB) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae318828-d4fb-4956-bcba-06eb7c27b302%2F7444b051-1ee6-4d71-b34a-c06941601a8d%2Fkh28po4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A given by
1. M₁1
4. M41
12. A =
14. A =
16. A =
A =
2-1 3 1
4 13-1
624 1
2 20-2
2. M₂1
5. M34
12 4
23 7
4 2 10
12 1
032
-1 1 1
200
310
24 2
3. M31
6. M43
13. A =
15. A =
17. A =
(7)
2-3 2
1
-1 -2
3
200
132
214
1215
0300
0412
0314
In Exercises 8-19, calculate the determinant of the given
matrix. Use Theorem 3 to state whether the matrix is
singular or nonsingular.
31]
-1 2
2 3
46
8. A =
10. A =
9. A =
11. A =
1 -1
-2
21
2
1
1
21
4.2 Determinants and the Eigenvalue Problem
289
23. Let A = (aij) be the (n × n) matrix specified thus:
aij = d for i = j and a¡j = 1 for i # j. For n = 2,
3, and 4, show that
det(A) = (d 1)"-¹ (d - 1 + n).
24. Let A and B be (n x n) matrices. Use Theorems 2
and 3 to give a quick proof of each of the following.
a) If either A or B is singular, then AB is singular.
b) If AB is singular, then either A or B is singular.
25. Suppose that A is an (n × n) nonsingular matrix,
and recall that det (I) = 1, where I is the (n x n)
identity matrix. Show that det(A-¹) = 1/det(A).
26. If A and B are (n × n) matrices, then usually AB ‡
BA. Nonetheless, argue that always det(AB) =
Transcribed Image Text:THEOREM 3
Let A be an (n x n) matrix. Then
A is singular if and only if det (A) = 0.
Theorem 3 is already familiar for the case in which A is a (2 x 2) matrix (recall
Definition 1 and Examples 1 and 2). An outline for the proof of Theorem 3 is given in
the next section. Finally, in Section 4.4 we will be able to use Theorem 3 to devise a
procedure for solving the eigenvalue problem.
We conclude this brief introduction to determinants by observing that it is easy to
calculate the determinant of a triangular matrix.
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