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Midterm

# MATH240 BOYLE-M FALL2012 0101 MID EXAM 2Exam

Department
Mathematics
Course Code
MATH 240
Professor
All
Study Guide
Midterm

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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.
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
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.