ques 6plz provide answer for ques 6 6.Read the following carefully, and then answer the questions that follow.We say that two n x n matrices are cousins, if there is some invertible matrix P so thatA = PBP-1"Notice that if A and B are cousins, then B = P-AP = (P|A(P)-1. Thus, the definition is "symmetric" (ie. the order we say itdoesn't matterk if A and B are cosusins then B and A are cousins.One of the first things you might notice about the notion of cousins, it that it is clearly connected todiagonalizability:Theorem 1A matrix A is diagonalizable if and only if there is a diagonal matrix D so that A and D arecousins.One might think from the definition that the main use of the definition of cousin matrices is toreframe diagonalizability. However, it actually allows us to generalize it in useful ways - in thefollowing example and the first question below, we'll see matrices A and B which are cousins,but where neither of which is diagonal, and for which techniques from our study of diagonalizablematrices can be applied.-7 -5]Let A =and let B = R3/2 =be the matrix of rotation by 3n/2.10-1 01Then with P=-12,if you compute PBP1, you will see that the end result is A.Thus, A and B are cousins.The next theorem shows us that cousins share a lot of important properties in common:Theorem 2If A and B are cousins, then:1. det(A) = det(B), and2. CA(x) = cB(x) (and hence A and B have the same eigenvalues.)Proof: Let A = PBP- for some invertible matrix P.1. det(A) = det(PBP-1) = det(P) - det(B) - det(P1) = det(B) (using that det(P) = 1/det(P1).)2. CA(x) = det(xi – A) = det(xI – PBP-1). But we can rewrite I as PIP-1 and substitutethis in; so, continuing we have: .. = det(XPIP-1- PBP-1).But then you can check that XPIP-1 - PBP-1 = P(xI – B)P1, by multiplying out theright-hand side. This allows us to continue with:... = det (P(xl – B)P1) = det(P) det (xl – B) det(P1) = det(xl - B) = cB(x)Continues on the next page..MAT223 - Winter 2022Midterm 26.1Use the result of the Example above to compute A100 (for A the matrix from the Example).Hint: consider (PBP-)100, What is Bl0026.2 () True or False: If A and B are nxn matrices so that det(A) = det(B), then A and B arecousins. Hint: part of Theorem 2 should be useful.63 ****) True or False: If A and B are cousins and v is an eigenvector of A, then v is aneigenvector of B.

A and B are cousins then they are diagonalizable.

Read the following carefully, and then answer the questions that follow. We say that two n x n matrices are cousins, if there is some invertible matrix P so that A = PBP1 *Notice that if A and B are cousins, then 5 = PAP = (P-A(P)-. Thus, the definition is "symmetric" (ie the order we say it doesn't matter): if A and B are cousins then B and A are counins. One of the first things you might notice about the notion of cousins, it that it is clearly connected to diagonalizability: Theorem 1 A matrix A is diagonalizable if and only if there is a diagonal matrix D so that A and D are cousins. One might think from the definition that the main use of the definition of cousin matrices is to reframe díagonalizability. However, it actually allows us to generalize it in useful ways - in the following example and the first question below, we'll see matrices A and B which are cousins, but where neither of which is diagonal, and for which techniques from our study of diagonalizable Thm Definition

Read the following carefully, and then answer the questions that follow. We say that two n x n matrices are cousins, if there is some invertible matrix P so that A = PBP1 *Notice that if A and B are cousins, then 5 = PAP = (P-A(P)-. Thus, the definition is "symmetric" (ie the order we say it doesn't matter): if A and B are cousins then B and A are counins. One of the first things you might notice about the notion of cousins, it that it is clearly connected to diagonalizability: Theorem 1 A matrix A is diagonalizable if and only if there is a diagonal matrix D so that A and D are cousins. One might think from the definition that the main use of the definition of cousin matrices is to reframe díagonalizability. However, it actually allows us to generalize it in useful ways - in the following example and the first question below, we'll see matrices A and B which are cousins, but where neither of which is diagonal, and for which techniques from our study of diagonalizable Thm Definition

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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