Math 3113-005: Linear Systems & Eigenvalues/Vectors Quiz 7 - Prof. Darryl Cullough, Quizzes of Mathematics

Download Math 3113-005: Linear Systems & Eigenvalues/Vectors Quiz 7 - Prof. Darryl Cullough and more Quizzes Mathematics in PDF only on Docsity! Mathematics 3113-005 Quiz 7 Form A April 15, 2011 Name (please print) Instructions: Give concise answers, but clearly indicate your reasoning. I. (3) The system of linear equations c1 + c2 = 10 c1 − c2 − c3 = −1 c1 + c3 = 12 arises in solving one of the homework problems (5.1#26). Use the method of Gauss-Jordan elimination to solve this system. That is, rewrite the system as an “augmented” matrix, then do elementary row operations to obtain the values of c1, c2, and c3 that satisfy the system. The first step, writing the augmented matrix, has already been carried out below, just continue the process from there.[ 1 1 0 10 1 −1 −1 −1 1 0 1 12 ] −→ II. (3) For the system X ′ = [ 0 1 1 1 0 1 1 1 0 ] X, a general solution is [ x1 x2 x3 ] = c1e −t [ 1 0 −1 ] + c2e −t [ 0 1 −1 ] + c3e 2t [ 1 1 1 ] (do not derive this or check this). Write a system of linear equations whose solution (c1, c2, c3) gives x1, x2, and x3 satisfying x1(1) = 0, x2(1) = −1, x3(1) = 7. Do not solve this system, just write it down. III. (3) Define an eigenvalue of a matrix A, and define an eigenvector associated to that eigenvalue. You may use the version of the definitions given in class, or the version given in the book, or any equivalent statement. IV. (7) (a) Show how to calculate that the eigenvalues of the matrix P = [ 3 4 3 2 ] are −1 and 6. (b) An eigenvector associated to the eigenvalue −1 is [ −1 1 ] (do not calculate this or check it). Use this to write out a solution X1 of the system X ′ = PX. (c) For the eigenvalue 6, find an associated eigenvector [ a b ] .