APPM 3310: Matrix Methods — Exam #1 — February 17, 2010
On the front of your bluebook write (1) your name, (2) “TEST 1/3310”, (3) “SPRING 2010”
and a grading table. A correct answer with no supporting work may receive no credit while an
incorrect answer with some correct work may receive partial credit. Start each problem on a
new page. No electronic devices of any kind (e.g. cell phones, calculators, etc.) are permitted. A
one-page sheet of notes is allowed. SHOW ALL WORK. JUSTIFY ALL YOUR ANSWERS.
Problem 1. (20 points) For this problem, let A=
1 0 −1
−3 1 0
2 3 −9
.
(a) Find the LU-decomposition of Awhere Uis in row echelon form.
(b) What is the rank of A?.
(c) Is Ainvertible? Why or why not?
(d) Find the determinant of A−T.
Problem 2. (20 points) Consider a linear system whose augmented matrix is of the form
1 1 3 2
1 2 4 3
1 3 a b
(a) For what values of aand bwill the system have infinitely many solutions?
(b) For what values of aand bwill the system be inconsistent?
(c) Suppose a= 5 and b= 4, solve the system.
Problem 3. (20 points) Let P(2) denote the vector space of all polynomials p(x) of degree less
than or equal to 2.
(a) Show that p1(x) = 1 + x2,p2(x) = x+x2and p3(x) = 1 + 2x+x2form a basis for P(2).
(b) Find the coordinates of f(x) = 1 + 4x+ 7x2in terms of the ordered basis given in part (a).
Problem 4. (20 points) Consider the vector space C2(R) consisting of all real valued functions
with domain Rthat have two continuous derivatives. Let Sbe the set of all functions y=f(x)
in C2(R) such that y00 +y0+y= 0. Is Sa vector space? Why or why not? Justify your answer
mathematically.
Problem 5. (20 points) Give a brief answer to each question. Show all work.
(a) Suppose Ais an n×nmatrix that satisfies A2−3A−I=O, then is Ainvertible? Why
or why not? If possible, find A−1.
(b) Suppose Ais a symmetric n×nmatrix. Show that A2−3A−Iis symmetric. Justify your
answer.
END