Homework Help is Here – Start Your Trial Now!
Math
Algebra
e. If AT-A, then det A = -1.

e. If AT-A, then det A = -1.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
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,...
See similar textbooks
icon
Related questions
Question
Please answer Exercise 3.2.20 e).
168 Determinants and Diagonalization Exercise 3.2.19 For each of the matrices in Exercise 2, find the inverse for those values of c for which it exists. Exercise 3.2.20 In each case either prove the statement or give an example showing that it is false: a. If adj A exists, then A is invertible. b. If A is invertible and adj A = A-¹, then det A = 1. c. det (AB) = det (BTA). d. If det A 0 and AB = AC, then B = C. e. If AT-A, then det A = -1. f. If adj A = 0, then A = 0. g. If A is invertible, then adj A is invertible. h. If A has a row of zeros, so also does adj A. i. det (ATA) > 0 for all square matrices A. j. det (I+A) = 1 + det A. k. If AB is invertible, then A and B are invertible. 1. If det A = 1, then adj A = A. m. If A is invertible and det A = d, then adj A = dA-¹. Exercise 3.2.21 If A is 2 x 2 and det A = 0, show that one column of A is a scalar multiple of the other. [Hint: Definition 2.5 and Part (2) of Theorem 2.4.5.] Lucanoico 2222 Dind a. (0, 1), b. (0, 1), c. (0, 2), Exercise 3.2. det A = 1+ a and c. Exercise 3.2.2 a. Show th only if c b. Show th ible, the Exercise 3.2.27 are integers. SI implies the oth- 1. A is inve 2. det A =
Transcribed Image Text:168 Determinants and Diagonalization Exercise 3.2.19 For each of the matrices in Exercise 2, find the inverse for those values of c for which it exists. Exercise 3.2.20 In each case either prove the statement or give an example showing that it is false: a. If adj A exists, then A is invertible. b. If A is invertible and adj A = A-¹, then det A = 1. c. det (AB) = det (BTA). d. If det A 0 and AB = AC, then B = C. e. If AT-A, then det A = -1. f. If adj A = 0, then A = 0. g. If A is invertible, then adj A is invertible. h. If A has a row of zeros, so also does adj A. i. det (ATA) > 0 for all square matrices A. j. det (I+A) = 1 + det A. k. If AB is invertible, then A and B are invertible. 1. If det A = 1, then adj A = A. m. If A is invertible and det A = d, then adj A = dA-¹. Exercise 3.2.21 If A is 2 x 2 and det A = 0, show that one column of A is a scalar multiple of the other. [Hint: Definition 2.5 and Part (2) of Theorem 2.4.5.] Lucanoico 2222 Dind a. (0, 1), b. (0, 1), c. (0, 2), Exercise 3.2. det A = 1+ a and c. Exercise 3.2.2 a. Show th only if c b. Show th ible, the Exercise 3.2.27 are integers. SI implies the oth- 1. A is inve 2. det A =
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS