**Problem 12****§7.3, Exercise 4.** Let $ A $ and $ B $ be similar $ n \times n $ matrices.(a) Show that if $ A $ is invertible, then $ B $ is invertible.(b) Show that $ A + A^{-1} $ is similar to $ B + B^{-1} $.**Explanation:** This problem explores the properties of similar matrices. If two matrices $ A $ and $ B $ are similar, there exists an invertible matrix $ P $ such that $ B = P^{-1}AP $. The exercises involve proving statements about invertibility and similarity transformations involving inverses.

Given: A and B are similar n×n matrices. To prove: a)Given: A is invertible. Since A and B are…

Answered: 87.3, Exercise 4. Let A and B be…

87.3, Exercise 4. Let A and B be similar n x n matrices. (a) Show that if A is invertible, then B is invertible. (b) Show that A+ A-1 is similar to B+B-!.

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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Concept explainers

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

Question

**Problem 12**

**§7.3, Exercise 4.** Let $ A $ and $ B $ be similar $ n \times n $ matrices.

(a) Show that if $ A $ is invertible, then $ B $ is invertible.

(b) Show that $ A + A^{-1} $ is similar to $ B + B^{-1} $.

**Explanation:**
This problem explores the properties of similar matrices. If two matrices $ A $ and $ B $ are similar, there exists an invertible matrix $ P $ such that $ B = P^{-1}AP $. The exercises involve proving statements about invertibility and similarity transformations involving inverses.

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