Quiz 6 in Mathematics 3113-005 - April 8, 2011 - Prof. Darryl Cullough, Quizzes of Mathematics

Download Quiz 6 in Mathematics 3113-005 - April 8, 2011 - Prof. Darryl Cullough and more Quizzes Mathematics in PDF only on Docsity! Mathematics 3113-005 Quiz 6 Form B April 8, 2011 Name (please print) Instructions: Give concise answers, but clearly indicate your reasoning. I. (5) Let A =  4t −1 0 2 1 −t 1 2 0 , B =  3 −1 0 2− t 1 1 , and C = [− cos(t) 3 0 ] . (a) Tell which of the following six products are defined (do not do any calculations, just tell which ones are defined): AB, BA, AC, CA, BC, CB. (b) Calculate det(A). II. (2) Define what it means to say that a collection of vectors {X1, X2, . . . , Xn} is linearly independent. III. (3) Write the system x′1 = 8x1 + tx2 + cos(t), x ′ 2 = x2 − x3, x′3 = t + 2tx2 − x3 in matrix form X ′ = PX + F . Do not proceed further with solving the system, just rewrite the general form X ′ = PX + F with X, P and F written as matrices with the correct dimensions and entries for this particular system. IV. (3) Write the second-order system x′′ − 2x + y = 0, y′′ + 2x − 3y = 0 as an equivalent system of first-order equations. V. (2) For the system X ′ =  0 1 1 1 0 1 1 1 0 X, verify that X = e−t  1 0 −1  is a solution. VI. (3) Bonus problem: Graph the hyperbola x2 − y 2 2 = 1, showing the numerical values of the intercepts and the equations of the asymptotes.