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Math / Linear Algebra: A Geometric Approach 2 / Chapter 4.3 / Problem 20

Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams

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Prove or give a counterexample: a. If B is similar to A, then BT is similar to AT. b. If

Chapter 4, Problem 20

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QUESTION:

Prove or give a counterexample: a. If B is similar to A, then BT is similar to AT. b. If B2 is similar to A2, then B is similar to A. c. If B is similar to A and A is nonsingular, then B is nonsingular. d. If B is similar to A and A is symmetric, then B is symmetric. e. If B is similar to A, then N(B) = N(A). f. If B is similar to A, then rank(B) = rank(A).

Questions & Answers

QUESTION:

Prove or give a counterexample: a. If B is similar to A, then BT is similar to AT. b. If B2 is similar to A2, then B is similar to A. c. If B is similar to A and A is nonsingular, then B is nonsingular. d. If B is similar to A and A is symmetric, then B is symmetric. e. If B is similar to A, then N(B) = N(A). f. If B is similar to A, then rank(B) = rank(A).

ANSWER:

Problem 20

Prove or give a counterexample: -

a. If is similar to , then is similar to

b. If is similar to , then is similar to

c. If is similar to and A is non-singular, then B is non-singular

d. If is similar to and A is symmetric, then B is symmetric

e. If is similar to , then

f. If is similar to , then

Step by Step Solution

Step 1 of 6

Given is similar to . Then, there is an invertible matrix P such that .

Now, taking transpose of both sides

Hence, is similar to .

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