Let p(x) be the polynomial The companion matrix of p(x) is the n x n matrix...
1. For a polynomial p(1) = cktk + Ck-14k-1 +...+ci+co, and an n x n matrix A, we define p(A) = CkAk + Ck-1 Ak-1 + ... +CjA + col. Let A be an n x n diagonalizable matrix with characteristic polynomial PA(1) = (1-2)*(1-3)n-k where 1 <k<n - 1. In other words, let A be an n x n diagonal- izable matrix that has only 2 and 3 as eigenvalues. Explain what is wrong with the following false "proof...
Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) = +-12+} Find the trace and determinant of A. 2 e: tr(4) and det(A) = 0 12: tr(A) = 0 and det(A) 2 3 2 T: tr(A) = 0 and det(A) 3 : None of the other answers 01 OW
Let A be an n × n matrix with characteristic polynomial f(t)=(−1)nt n + an−1t n−1 + ··· + a1t + a0. (a) Prove that A is invertible if and only if a0 = 0. (b) Prove that if A is invertible, then A−1 = (−1/a0)[(−1)nAn−1 + an−1An−2 + ··· + a1In]. 324 Chap. 5 Diagonalization (c) Use (b) to compute A−1 for A = ⎛ ⎝ 12 1 02 3 0 0 −1 ⎞ ⎠ . #18 a, b...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n. Problem 4 Let V be the vector space of functions of...
- © consider the companion form matrix a) b) show its characteristic polynomial is Pa () = 1 +2, ++24+24 show if a; is an eigenvalue of A, then (x 1; 1; 1) is the corresparling eigenvector.
Throughout this question, fix A as an n×n matrix. If f(x) is a polynomial, then f(A) is the expression formed by replacing every x in f(x) with A and inserting the n×n identity matrix I to its constant term. For example, if f(x) = x2 −2x+5 (whose degree is 2), then f(A) = A2 −2A+5I; if f(x) = −x3 +2 (whose degree is 3), then f(A) = −A3 + 2I. (a) Using induction of the degree of the polynomial f(x),...
Q1. Let A = be a 2 x 2 matrix. 30 (a) Find the characteristic polynomial of the matrix A. (5 pts) (b) Find all eigenvalues and associated eigenvectors of the matrix A. (10 pts) (c) If is an eigenvalue of A, what do you think it would be the eigenvalue of the matrix 7A?(Justify your answer) (5 pts)
Review 4: question 1 Let A be an n x n matrix. Which of the below is not true? A. A scalar 2 is an eigenvalue of A if and only if (A - 11) is not invertible. B. A non-zero vector x is an eigenvector corresponding to an eigenvalue if and only if x is a solution of the matrix equation (A-11)x= 0. C. To find all eigenvalues of A, we solve the characteristic equation det(A-2) = 0. D)....
Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for 3 x 3 determinants. [Note: Finding the characteristic polynomial of a 3 x 3 matrix is not easy to do with just row operations, because the variable is involved.] 4 0 | 4 8 1 -2 2 0 -3 The characteristic polynomial is (Type an expression using , as the variable.) Find the characteristic polynomial of the matrix, using either a cofactor expansion...
(1 point) Find the characteristic polynomial of the matrix -2-20 1 -1 0 p(x)