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**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 U−1AU =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.

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