2 0 0 2. Let A be the diagonal matrix 0 4 0First read Exercise 2 of Section 1.5, before continung (a) What would it mean to say that A is nonsingular? (b) Prove that A nonsingular. Give a full explanation using your definition in part Let A be a 4 × 4 matrix with its third row consisting of zeros. (a) What would it mean to say that A is nonsingular? (b) Prove that A is singular. (Hint: Exercise...
For a matrix A 3 5 (a) Find a nonsingular matrix Q let D-AQ be a diagonal matrix b)Find the inverse of A 氿ㄧ '11.ril. IfA+小1, find the maximal and minin al values of毗孵. 2 For a matrix A 3 5 (a) Find a nonsingular matrix Q let D-AQ be a diagonal matrix b)Find the inverse of A 氿ㄧ '11.ril. IfA+小1, find the maximal and minin al values of毗孵. 2
Suppose A and B are matrices with matrix product AB. If bi, b2, ..., br are the columns of B, then Ab, Ab2, ..., Ab, are the columns of AB 1. Suppose A is an nxnmatrix such that A -SDS where D diag(di,d2,... dn) is a diagonal matrix, and S is an invertible matrix. Prove that the columns of S are eigenvectors of A with corresponding eigenvalues being the diagonal entries of D Before proving this, work through the following...
Exercise 5 Let 2 1 -2 -1 (a) Find A4 (b) Find eA (c) Find a matrix B such that B2 = A. Exercise 5 Let 2 1 -2 -1 (a) Find A4 (b) Find eA (c) Find a matrix B such that B2 = 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 2: Let 4 1 2 5 1-1 0 3 2 0 3 2 a) Find the eigenvalues, eigenspaces of the linear operators LB, Lo. b) Using part a), find a basis for R3 that diagonalizes the linear operators c) Write B- EDE- with D a diagonal matrix. d) Find the eigenvalues, eigenspaces, and generalized eigenspaces of LA Problem 2: Let 4 1 2 5 1-1 0 3 2 0 3 2 a) Find the eigenvalues, eigenspaces of the linear...
linear algebra Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
Let matrix M = -8 -24 -12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP^−1. If not, explain carefully why not.
Let matrix M = -8 -24 12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP−1. If not, explain carefully why not.
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.