APPM 2360 Exam 2 Summer 2008
ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your student ID number, (3)
your lecture section, (4) your instructor’s name and (5) a grading table. You have 90 minutes to
work all 5 problems on the exam. Each problem is worth 20 points. Show ALL of your work in the
bluebook and box in final answers. Start each problem on a new page. A correct answer with no
relevant work may receive no credit, while an incorrect answer accompanied by some correct work
may receive partial credit. Text books, class notes and calculators are NOT permitted. One letter
size (8.5′′ ×11′′) crib sheet with anything hand written on both sides is allowed.
1. a) Given the matrices A=
1 2
2 3
3 4
and B=−1 0 −3
0 4 4 ,which of the following
matrix operations are defined and which are not defined? (You do not need to give
reasons or evaluate those that are defined.)
i) AB
ii) A2
iii) ATB
iv) BBT
v) A+BT
b) Determine all values of aand bfor which the matrix C=3a
0bhas only one eigen-
vector.
c) Give an example of three 2 ×2 matrices D,Eand F, with E6=Fbut DE =DF .
d) Prove there does not exist a matrix G, with G2=2−3
6−1and G3=−2−5
8−7.
(hint: think determinants)
2. Consider the matrix, A(t) = cos(t) sin(t)
−sin(t) cos(t).
a) Compute the determinant of A(t).
b) Use your answer from part (a) to answer the following. Give reasons.
i) Are cos(t) and sin(t) linearly independent functions?
ii) Are there any values of tfor which the columns of A(t) do not span R2?
c) Compute the eigenvalues of A(t).
d) Compute the eigenvectors of A(π
2).
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