APPM 3310: Matrix Methods — Exam #1 — February 27, 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. A correct answer with no supporting work may receive no
credit while an incorrect answer with some correct work may receive partial credit. Textbooks, class
notes and calculators are not permitted.
Please sign your bluebook under the Honor Code to indicate that you have neither given
nor received unauthorized assistance on this exam.
1. (50 points). Each answer for the following statements is True or False. Your answer must be
justified. If the answer is True then you must explain why it is True; if it is False you must give
a counterexample. Assume all matrices in this problem are square.
(a) For two matrices Aand B, if AB =0then A=0or B=0.
(b) If Ais a nonsingular matrix then AB =0implies that B=0.
(c) If Cand Dare symmetric matrices, so is CD.
(d) If A=LU, then det(A) = det(U).
(e) If Ais a 3 ×3 matrix and the equation Ax= (1,0,0)Thas a unique solution, then Ais
invertible.
2. (20 points) Let Vbe the set of polynomials in P(3) that satisfy p(1) = 0 and p(−1) = 0. That
is, V={p(x) = ax3+bx2+cx +d|p(−1) = p(1) = 0}
(a) Show that Vis a subspace of P(3) . (A complete answer for this question will include a
definition of subspace.)
(b) Find a basis for V.
3. (40 points) For this problem, use the matrix A=
1 0 0 3
2 2 −1 2
3 2 −1 5
(a) Give the definition for the kernel of an arbitrary m×nmatrix. Find a basis for ker Afor
Agiven in this problem.
(b) Give the definition for the range of an arbitrary m×nmatrix. Find a basis for rng(A).
(c) State the Fundamental Theorem of Linear Algebra. Give the dimensions of each of the four
fundamental subspaces of matrix A.
(d) For what values of kdoes the system Ax=bfor b= (1, k, 2)T, have a solution? Find the
general solution(s) for these values of k.