Q.12. WHICH OF THE GIVEN BELOW ARE ALWAYS TRUE? MAY BE MORE THANONE RIGHT ANSWER1.If A is diagonalizable then cA is diagonalizable for anyconstant of c.2.If the characteristic polynomial of a diagonalizable matrix A is equal to ((A –7)^(3))((A – 5)^(6))((A – 8)^(2)) then the eigenvalue A = 7 has a geometricmultiplicity of 3.3.If A is similar to B, then AT is similarto B.

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Answered: Q.12. WHICH OF THE GIVEN BELOW ARE…

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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in 2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 in , then define p (A) = E a¡A' . If, is an eigenvalue of A, the show that p (1) is an eigenvalue of p(A). i = 1
Which of the following statements are always true? (i) If A is a 3 x 3 diagonalizable matrix then A has 3 distinct eigenvalues. (ii) If the characteristic polynomial of a matrix A is equal to (2 – 3)*(2– 6)°(2 – 7)š then the eigenvalue 2 = 3 has a geometric multiplicity of 4. (iii) If A is similar to B, and B is symmetric, then A' is similar to B. A) (i) and (ii) only (B) (ii) and (iii) only (C) (ii) only (D) none of them (E) (iii) only (F) (i) only G) all of them (H) (i) and (iii) only
1. Let A = has eigenvalues X₁ = 2 2 3 -1 -2 2 1 and X₂ 5. You can assume, and do NOT need to show, that A Compute algebraic and geometric multiplicities of X₁ and ₂.
11. Let (b) (c) (d) 1 1 3 3 -3 -5 -3 3 3 1 Find the eigenvalues of A. (Hint: X = -2 is an eigenvalue with multiplicity 2.) A = Find the eigenvectors of A. Find matrices P and such that A = PDP-¹, Use the information above to find A4.
3. Consider the following matrix and answer the following questions: -CO A = (a) 12 points Without doing any calculations explain why A is diagonalizable. A is diagnozible because it's olso is invertible Compute the eigenvalues of A. (5² (c) 16 points) Compute the eigenvectors of A. -1- AC
2. Find the characteristic polynomial, eigenvalues, eigenvectors, and (if possible) an invertible matrix P such that P-¹BP is diagonal where B = 1 2 1 -2 3 6 -6 2
None
2.55 Find the eigenvalues and eigenvectors of the following matrices: 2√3 01 4 0 04 (a) (c) 4 -4 0 0 -4 0 0 0 3 1 0 0 0 3 -1 0 -1 3 (b) (d) [ 0 2 -1 1 1 0 1 1 1 2
1. For each of the following statements, check if it is True or False. (a) True or False: For each 2 x 2 mtrix A, there exists a matrix B such that det(A+B) + det(A) + det(B). (b) True or False: Let A be a 3 x 3 matrix with 3 distinct eigenvalues. Then there is an eigenbasis for A. (c) True or False: Let A be a n x n matrix with one eigenvalue A = 0. Then Ax infinitely many solutions for any b e R". b has %3| (d) True/False: An n x n matrix A is similar to a matrix B if and only if |A| = |B|.
Let 1 3 4. A = -5 -3 and B = -4 -6 -3 3 1 3 1 For this problem, you may use the fact that both matrices have the same characteristic polynomial: PA(A) = PB(A) = -(A – 1)(A+2)². (a) Find all eigenvectors of A. (b) Find all eigenvectors of B. (C) Which matrix A or Bis diagonalizable? (d) Diagonalize the matrix stated in (C), i.e., find an invertible matrix P and a diagonal matrix D such that A = PDP B= PDP 1. 1 or %3D %3D
For which value of k does the matrix 4 k A : 6 3 have one real eigenvalue of multiplicity 2? k =

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