Study Guides (380,000)
US (220,000)
UMD (10,000)
MATH (6,000)
MATH 240 (100)
All (100)
Midterm

MATH240 BOYLE-M FALL2012 0101 MID EXAM 2Exam


Department
Mathematics
Course Code
MATH 240
Professor
All
Study Guide
Midterm

This preview shows half of the first page. to view the full 2 pages of the document.
MATH 240 – Fall 2012 – Exam 3
CALCULATORS ARE NOT ALLOWED
CLOSED BOOK, NO NOTES
TURN OFF ALL ELECTRONIC DEVICES
There are 5 questions. Answer each on a separate sheet of paper. Use the back side if needed.
On each sheet, put your name, your section leader’s name and your section meeting time.
When a question has a short final answer, put a BOX around that answer.
NOTATION: If Ais a matrix, then ATdenotes the transpose of A.
1. (a) (20 points) Let A=5 7
6 6. Write down a matrix Uand a diagonal
matrix Dsuch that U1AU =D.
(b) (4 points) Give an example of a 2 ×2 matrix Bwhich has a real
eigenvalue and is not diagonalizable. No proof necessary.
2. (18 points) There is a change of variable, x=P y, that transforms the
quadratic form Q(x) = x2
1+ 10x1x2+x2
2into a quadratic form with no
cross term. (In other words, with x=P y, there are numbers α, β such
that Q(x) = αy2
1+βy2
2.)
(a) (4 pts.) What is the symmetric matrix Asuch that Q(x) = xTAx ?
(b) (6 pts.) What are the numbers αand β?
(c) (8 pts.) Produce an orthogonal matrix Pwhich gives the change of
variable.
THERE ARE MORE QUESTIONS ON THE OTHER SIDE
OF THIS PAPER.
You're Reading a Preview

Unlock to view full version