APPM 3310: Matrix Methods — Final Exam — May 9, 2006
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Show all work in your bluebook. Textbooks, class notes and calculators are not permitted,
although you are allowed to use one page of notes as a reminder sheet. If you find that the arithmetic
for this exam seems complicated, go back and check your work.
Please sign your bluebook under the Honor Code to indicate that you have neither given
nor received unauthorized assistance on this exam.
Please skip one of the 30 point problems, #2, 4, or 5. Write the number of the skipped
problem on the front of your bluebook.
1. (50 points) Each of the following unrelated questions involves a concept from class. The concept
is in bold-face. A complete answer for each part will include a definition of the bold-faced word.
Then, use your definition in your answer to the question.
(a) Are the 4 matrices given by A1=1 0
0 1 ,A2=1 1
0−1,A3=2 0
0 1 ,A4=
0 0
1 0 , a basis of M2×2?
(b) What are the L1,L2, and L∞norms on [−1,1] for the function f(x) = x−(1/4)?
(c) Find the rank of the m×nmatrix A=vwT, where vis a nonzero m×1 vector and wTis
a nonzero 1 ×nrow vector.
(d) Is the function F: IR2→IR2given by Fx
y=xy
x−yalinear function? Explain.
2. (30 points) For this problem, let A=
1 0 −1
0 1 −2
1−2 3
.
(a) Find corng(A) and ker(A).
(b) Find conditions on vector bso that Ax =bhas a solution.
(c) For b=
3
1
1
find the general solution Ax =bgiven by x=w+zwith w∈corng(A) and
z∈ker(A).
3. (35 points) As you know, an n×nmatrix is symmetric if A=AT. An n×nmatrix Jis called
skew-symmetric if J=−JT. (Skew-symmetric matrices arise in many applications, particularly
in physics.)
(a) Suppose a matrix Jis skew-symmetric. What can you say about the diagonal entries of J,
Jii for i= 1, . . . , n? Explain.
(b) Write down an example of a 2 ×2, non-zero, skew-symmetric matrix.
(c) Does the set of 2 ×2 skew-symmetric matrices form a subspace of the vector space of all
2×2 matrices, M2×2? Explain fully, using the definition of a subspace in your answer.
(d) For your example in part (b) show that vTJv = 0 for all v∈
R
2.
(e) Show that vTJ v = 0 for all v∈
R
n, for any n×n, skew-symmetric matrix J.
