MATH 240 Midterm: MATH240 BOYLE-M FALL2012 0101 MID SOL 2

• School: University of Maryland
• Department: Mathematics
• Course: MATH 240
• Professor: All

Document Summary

Math 240 fall 2012 exam 3 solutions. Notation: if a is a matrix, then at denotes the transpose of a: (a) (20 points) let a = (cid:18)5 7. Write down a matrix u and a diagonal matrix d such that u 1au = d. The eigenvalues of a are 12 and -1. We nd eigenvectors for these and use then as columns of u : 0 1(cid:19) . (b) (4 points) give an example of a 2 2 matrix b which has a real eigenvalue and is not diagonalizable. 0 1(cid:19) . (cid:18)1 1: (18 points) there is a change of variable, x = p y, that transforms the quadratic form q(x) = x2. 2 into a quadratic form with no cross term. (in other words, with x = p y, there are numbers , such that q(x) = y2. What is the symmetric matrix a such that q(x) = xt ax ? (b) (6 pts. )