Let T: V → V be a linear operator whose characteristic polynomial is pT(x)=x2(x−1)3(x+2)2. Judge each item below as true or false. (A) Let A be the matrix of T with respect to some basis of V. If A(A−I)(A+2I) = 0, then T is diagonalizable. (B) The roots of the minimal polynomial of T are 1 and −2. (C) If the minimal polynomial of T is x(x−1)(x+2) then there is a basis of V formed by eigenvectors.

Let T: V → V be a linear operator whose characteristic polynomial is pT(x)=x2(x−1)3(x+2)2. Judge each item below as true or false. (A) Let A be the matrix of T with respect to some basis of V. If A(A−I)(A+2I) = 0, then T is diagonalizable. (B) The roots of the minimal polynomial of T are 1 and −2. (C) If the minimal polynomial of T is x(x−1)(x+2) then there is a basis of V formed by eigenvectors.

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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Question

Let T: V → V be a linear operator whose characteristic polynomial is pT(x)=x2(x−1)³(x+2)². Judge each item below as true or false.

(A) Let A be the matrix of T with respect to some basis of V. If A(A−I)(A+2I) = 0, then T is diagonalizable.
(B) The roots of the minimal polynomial of T are 1 and −2.
(C) If the minimal polynomial of T is x(x−1)(x+2) then there is a basis of V formed by eigenvectors.

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