APPM 3310: Matrix Methods — Final Exam — December 12, 2005
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 a one page 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.
1. (20 points) Here are some short answer questions. As always, explain your answer!
(a) Suppose Ais a 3 ×3 matrix with diagonal elements 1,3,and −4, and that two of its
eigenvalues are λ1= 2 and λ2= 3. What is the third eigenvalue?
(b) If Ais a square matrix and A= 5B, then the eigenvalues of Aare five times the eigenvalues
of B. True or False?
(c) Prove that the product of the singular values of a square, nonsingular matrix Ais equal to
|det A|.
(d) Suppose a 3 ×3 matrix Ahas eigenvalues λ= 1,2,and 3. What are the eigenvalues of
B=A−2I?
2. (30 points) Consider the system Ax=b, with A=
2 2
2 1
−1−1
and b=
1
3
2
.
(a) Find a basis for rng(A).
(b) Find a basis for coker(A). Explain how your calculation is consistent with the fundamental
theorem of Linear Algebra.
(c) Does the system Ax=bhave a solution? Why or why not?
(d) Find the least squares solution to Ax=b.
3. (20 Points) A few True-False questions. You must explain your answer!
(a) The quadratic form q(x, y) = x2+ 2xy + 3y2is positive definite.
(b) The expression hv,wi=v1w1+ 2v1w2+ 3w2v2is an inner product on R2.
(c) If AB =I, then BA =I.
(d) If Aand Bare invertible, then so is A+B.
(e) If vand ware nonzero column vectors in Rn, then rank(vwT) = 1.
