Please answer 3 In the Rabbit example we studied L R2 Righta

Please answer #3

In the Rabbit example, we studied L: R^2 Rightarrow R^2- where L([a, b]^T) = [a + b, 2a]^T. Find the matrix representation A of L (w.r.t. the std basis ofR^2). We found two eigenvectors of A. x_1 = [1, 1]^T and x_2 = [1, -2]^T, which form a basis X of R^2. Find the corresponding eigenvalues. Find the matrix representation 13 of L w.r.t. the basis X. Id) Find an eigenvector for B Is it also an eigenvector of A ? Explain briefly. Let x = [2, 3]^T_x (the coordinates are wrt the eigenvector basis, X). Find [L(x]_s, in standard coordinates. True-False. You can assume t he matrices are all square. A^T.A and A A^T always have the same rank. If A and B are unitary then AB is unitary. If A is singular then AB is singular. If A^H = - A then A is normal. If A is unitary then A is not defective. If A is Hermitian and unitary then A^2 = 1. If U. V are subspaces of R^n. and U Perpendicular V then V subset U^Perpendicular If A is unitary and x belongs to C^n then ||Ax|| = ||x||. If A is an eigenvalue of a unitary matrix then |Lambda| = 1. If A is an eigenvalue of a unitary matrix then Lambda is real Suppose A is a 4 Times 4 matrix and det .A = 3. Find det (adj(A)). Choose ONE of these to prove. If an n Times n matrix A is diagonalizable, then it has n L.I. eigenvectors. State and prove the Spectral Theorem. You are given this A = QR factorization to help with the questions below. Note that the1/2 scalar is part of Q.

Solution

3) we know that |adj(A)|=|A|^n-1

here n is order of matrix A

given A is 4X4 matrix

so n=4

given |A|=3

so |adj(A)|=3^4-1=3³=27