7. Let A be an 12 x 12 matrix given by21A21212-33 130(a) What are the eigenvalues of A?(b) What are algebraic and geometric multiplicities for each of the eigenvalues?(c) What is the characteristic polynomial of A?(d) What is trace(A)? What is det(A)? What is rank(A)?

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Answered: 7. Let A be an 12 x 12 matrix given by…

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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2. Without using Sage, find • the characteristic polynomial • the eigenvalues (include the multiplicity of each) • bases for the eigenspaces (indicate the dimension of the eigenspaces) of the following matrices: [30 (a) B= (b) C: = 4 (c) D = -2 10 (Once you've found the characteristic polynomial, you may find -20 its factorization by using Sage. For example, to factor x²-3x+2 in Sage: f = x^2-3*x+2 f.factor())
An n × n matrix A has the characteristic equation∣λI − A∣ = (λ + 2)(λ − 1)(λ − 3)2 = 0.(a) What are the eigenvalues of A?(b) What is the order of A? Explain.(c) Is λI − A singular? Explain.(d) Is A singular? Explain.
The characteristic polynomial of the matrix A is p(x) = (A-3)²1² The matrix has two real eigenvalues X₁
Suppose that A is a square matrix with characteristic polynomial (2-3)*(2-4)³ (2+1). (a) What are the dimensions of A? (Give n such that the dimensions are n x n.) n= (b) What are the eigenvalues of A? (Enter your answers as a comma-separated list.) λ = (c) Is A invertible? Yes O No (d) What is the largest possible dimension for an eigenspace of A?
Consider the matrix A = - 13 24 8 15 (a) Calculate the det(A - xI), where x is a variable. We call the result the characteristic polynomial of A. (b) Give the two zeros 1 and 2 of this polynomial. λ1 and 22 are called the eigenvalues of the matrix A. (c) For each eigenvalue 21, find an eigenvector vi = ( ) + (8) xi # such 7 that Avi = hivi. (d) Construct the matrix D = (2). are eigenvectors v1 and v2. Calculate QDQ λ1 0 22 0 and the matrix Q whose columns -1 ; what do you notice?
3 Let A = 4 -3 (a) Find the eigenvalues of and the corresponding eigenvectors for the matrix A. (b) Find an matrix P such that t P" AP diagonalizes A.
(1) (a) Diagonalize the matrices ($) () 2 -1 11 3 15 and 1 3 -1 -4 -1 -2 2 (The eigenvalues of the second matrix are 1 = 1, 5.) (b) Calculate lim A", where A is the first matrix.
(i) (; ) 8 4 4 and are eigenvectors of A 7 7 5 a Use the definition of eigenvalues and eigenvectors to find the eigenvalues of A. b Hence write down the characteristic polynomial of A.
Consider the matrix 0 1 0 0 0010 0001 1000 (a) Find the eigenvalues by finding the characteristic polynomial and looking at its roots. (b) Find the eigenvalues by setting up a system of equations Ar = λ and solving the easy system of equations in the variables x1, x2, x3, x4 for possible X's. (c) For (each/the) eigenvalue, find the corresponding eigenvectors and eigenspaces.
Consider matrix A (attached in image) ( a) Find the characteristic polynomial of matrix A.( b) Find eigenvalues of the matrix A.(c) Find a basis for the eigenspaces of matrix A.
43. Linear Transformation
Express the integral 11. RK f(x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. E y = x², z = 0, y + 2z = 4

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7. Let A be an 12 x 12 matrix given by 2 1 2 A = 1 2 1 2 -3 3 1 3 5 01 (a) What are the eigenvalues of A? (b) What are algebraic and geometric multiplicities for each of the eigenvalues? (c) What is the characteristic polynomial of A? (d) What is trace(A)? What is det(A)? What is rank(A)?