1. For a matrix A denote the characteristic polynomial of A by FA(t), i.e. Ea(t) =...
3. (a) For the following matrix A, compute the characteristic polynomial C(A) = det(A ?): A-1 1 (b) Find all eigenvalues of A, using the following additional information: This miatrix has exactly 2 eigenvalues. We denote these ??,A2, where ?1 < ?2. . Each Xi is an integer, and satisfies-2 < ?? 2. (c) Given an eigenvalue ?? of A, we define the corresponding eigenspace to be the nullspace of A-?,I; note that this consists of all eigenvectors corresponding to...
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
Q5. Consider the square matrix A 4 -3 2 (a) Show that the characteristic polynomial of A is: p(x) = 12 – 61 – 7. (b) Compute the matrix B= A2 – 6A – 712. (c) Show that A² – 6A = 712 for the given matrix A. (d) Is it possible to use the equation A2 – 6A = 712 to find the inverse of the given matrix A? (Justify your answer)
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...
6 4 4 1] 4 6 14 TP08: (10pts) The characteristic polynomial of the matrix A = is given by 4 1 6 4 4 4 6 p(t)= (1 + 1)(t – 5) (1 - 15). Find all eigenvalues of A. Is the matrix A defective?
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A).
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
use the definition of the characteristic polynomial for now on.
χA(λ) = det(λI − A).
1. Find an invertible matrix P and a diagonal matrix D such that A = PDP-1 where
Find the characteristic polynomial and the real eigenvalues of the matrix. | -5 -1 The characteristic polynomial is (Type an expression using , as the variable.) The real eigenvalues of the matrix are 7. (Use a comma to separate answers as needed.)
Find the characteristic polynomial and the eigenvalues of the matrix. 3 1 -15 The characteristic polynomial is (Type an expression using à as the variable. Type an exact answer, using radicals as needed.) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The real eigenvalue(s) of the matrix is/are (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.) OB. The...
Q5. Consider the square matrix A - 6 4 3 (a) Show that the characteristic polynomial of A# (X) = x-91-2. (6 pts) (b) Compute the matrix B-A 9A 21. (5 pts) (c) Show that A2 9A-21, for the given matrix A. (5 pts) (d) Is it possible to use the equation A? (Justify your answer) (5 pts) 9A 21, to incl the inverse of the given matrix A