1 MATH 21b Practice Questions Problem 1: Let A denote the matrix 1 41 !2" # $ % a) Find the eigenvalues and eigenvectors of A. b) Solve the dynamical system r x (m+1) = A r x (m) given that r x (0) = 23! " # $ . Thus, give r x (m) for m = 1, 2, 3, … . Problem 2: Which of the following is the equation for the best fit as determined by the least squares method for a line through the four points 03! " # $ ,13! " # $ , 14! " # $ , !11" # $ % in the x-y plane? Please justify your work. a) y = 1 41 1x + 2 72 2 b) y = 1 11 3x + 2 71 1 c) y = 1 31 1x + 2 11 1 d) y = 1 11 4x + 2 12 2 e) y = 1 31 1x + 2 71 1 f) y = 1 41 1x + 2 12 2 Problem 3: The vector 1!2" # $ % is the eigenvector with eigenvalue 3 of a symmetric 2 × 2 matrix with trace equal to 1. Write down the matrix.2 Problem 4: Circle T if the accompanying statement is true, and circle F if it is false. You need not justify your answers. T F a) There are infinitely many 2 × 2 matrices with determinant equal to 1 and trace equal to 2. T F b) All invertible matrices are diagonalisable. T F c) A non-zero matrix with 2 columns and 4 rows must have 2-dimensional image. T F d) SAS-1 is diagonal if S = 141 2!1 2" # $ % and if A is a 2 × 2 matrix with eigenvectors 21! " # $ and !21" # $ % . Answers: 1. a) 2, -3 with eigenvectors r e 1 = 41! " # $ and r e 2 = !11" # $ % b) 2m r e 1 + 2 (-3)m r e 2. 2. e) 3. !1 !2!2 2" # $ % 4. T, F, F,
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