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 ⎞ ⎠ .
Let A be an n × n matrix with characteristic polynomial f(t)=(−1)nt n + an−1t n−1...
1. Consider the matrix (a) Find the characteristic polynomial and eigenvalues of A (b) Find a basis for the eigenspace corresponding to each eigenvalue of A. (c) Find a diagonalization of A. That is, find an invertible matrix P and a diagonal matrix such that A - POP! (d) Use your diagonalization of A to compute A'. Simplify your answer.
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)-
linear algebra 1. Let A be a square matrix with characteristic polynomial 13 – 912 + 181 = 0. (a) What is the size of A? (b) Is A invertible? Why or why not? (C) How many cigenspaces does A have?
Problem 16 (10 pts) For an n x n matrix A, pa(t) = t.q(t) for some polynomial q(t) precisely when Det(A) = 0. Problem 17 (10 pts) If W CR” is a subspace and v eR”, then pw(v) is the least-squares approximation to v by a vector in W except when pw(v) = 0. Problem 18 (10 pts) If A is a real n xn matrix, then the pairing defined by <v, w >:=yT * AT * A* w is...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
42. Let (an) be the sequence defined by ao (0,Vn2 1, an+1 = sin(a,) T 1 1 (a) Show that lim nan (b) Deduce the nature of the series 3 1an 42. Let (an) be the sequence defined by ao (0,Vn2 1, an+1 = sin(a,) T 1 1 (a) Show that lim nan (b) Deduce the nature of the series 3 1an
let fx be a polynomial of degree <= to n whats the value of f(Xo, X1....Xn). explain Let f(x) = ao tai xt...... + Anxh be a polynomial of | degree less than or equal to n, and let {xo.xi... n} be distinct points What is the value of f[xo, X.. Xn] Justify / Explain.
4. Let T be a linear operator on the finite-dimensional space V with eharacteristie polynomial and minimal polynomial Let W be the null space of (T c) Elementary Canonical Forms Chap. 6 226 (a) Prove that W, is the set of all vector8 α in V such that (T-cd)"a-0 for some positive integer 'n (which may depend upon α). (b) Prove that the dimension of W, is di. (Hint: If T, is the operator induced on Wi by T, then...
3. Let A be the matrix 1 -2 (a) What is the characteristic polynomial for A? (b) What are the eigenvalues of A? (c) What are the eigenvectors of A?
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)....