Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) =...
Let p(x) be the polynomial The companion matrix of p(x) is the n x n matrix 1 1 n-2 .. -a-a0 cp) = 10 1 0 Find the companion matrix of p(x) - x3 + 5x2 - 2x 15 and then find the characteristic polynomial of C(p). C(p) det(C(p) Xr)-
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is the determinant of A. (b) If a 2 x 2 matrix satisfies tr(AP) = 5, tr(A) = 3, then find det(A). (The trace of a square matrix A, denoted by tr(A), is the sum of the elements on the main diagonal of A.
please do both 1 & 2 () There is interesting relationship2 between a matrix and its characteristic equation that we explore in this exercise. 2 (a) We first illustrate with an example. Let B - 1 -2 i. Show that 2-4 is the characteristic polynomial for B ii. Calculate B2. Then compute B2+ B 412. What do you get? (b) The first part of this exercise presents an example of a matrix that satisfies its own characteristic equation. Explain for...
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...
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...
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> Answer the following questions about the square matrix A of order 3: A= III. The square matrix B of order 3 is diagonalizable and meets AB=BA. prove that any eigenvector p of A is also an eigenvector of B. IV. Find the square matrix B of order 3 that meets B2 = A, where B is diagonalizable and all eigenvalues of B are positive. V. The square matrix X of order 3 is diagonalizable and meets AX =...
Problem #30: [2 marks] Suppose that a matrix A has characteristic polynomial p() = 1 - 31' + 814 - 23. Consider the following statements. (i) i = 2 is an eigenvalue of A. (ii) A is a 4 x 4 matrix. (iii) That same p() is also the characteristic polynomial of A! Determine which of the above statements are True (1) or False (2). So, for example, if you think that the answers, in the above order, are True...
3. Let Z= (3 a 2 x 2 matrix over Zs. Find the characteristic polynomial of Z and determine for which values of h e Z5, Z is diagonalizable.
Problem 2. (50 points) Suppose that a 4 x 4 matrix A with rows it. 73, 74, and has determinant det A = 1. Find the following determinants: ☺ U2 ü det 604 det det A +50 Note: You can earn partial creat on this problem Problem 1. (50 points) Let Rem -5 -20 -1 -1 - 1 4 16 0 الها (a) Compute dexA Use Cramer's rule to solve the following system 20x -5x -X -4 3 + 16x9...