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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 1 2 1 032 -1 1 1 200 3 1 0 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 ajj = 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 (nxn) 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 4 13-1 624 1 2 20-2 2. M₂1 5. M34 12 4 23 7 4 2 10 1 2 1 032 -1 1 1 200 3 1 0 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 ajj = 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 (nxn) matrices, then usually AB # BA. Nonetheless, argue that always det(AB) =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Linear algebra: please solve q14 and 17 correctly and handwritten. Theoram 3 is also attached 

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: ajj = 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 xn) 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 xn) matrices, then usually AB ‡ BA. Nonetheless, argue that always det (AB) =
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: ajj = 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 xn) 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 xn) matrices, then usually AB ‡ BA. Nonetheless, argue that always det (AB) =
THEOREM 3 Let A be an (n xn) 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.
Transcribed Image Text:THEOREM 3 Let A be an (n xn) 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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