APPM 3310: Matrix Methods — Exam #1 — September 28, 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. 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) For this problem, use the matrix A=
1 2 3
0 2 2
0−1−1
3 2 5
(a) Find the LU factorization of A,A=LU.
(b) Identify the pivot elements of U.
(c) What is rank(A)?
(d) Find the general solution to the system Ax=0.
(e) For what values of kdoes the system Ax=bfor b= (1,2,−1, k)T, have a solution? Find
the general solution(s) for these values of k.
(f) Give the definition for the kernel of an m×nmatrix. Find a basis for kerA.
(g) Give the definition for the range of an m×nmatrix. Find a basis for rng(A).
(h) What is dim(coker(A))?
(i) What is dim(corange(A))?
2. (50 points) A few unrelated, short answer questions.
(a) Give the definition for Wto be a subspace of a vector space V. Is the set of n×nmatrices
with det A= 0 a subspace of the vector space Mn×n? Explain.
(b) Are the polynomials p1=x2+ 1, p2= (x−1)2linearly independent? Do they span P(2)?
Explain.
(c) For which value(s) of kdoes the system
x+ky = 4
kx +y= 4
have (i) no solution, (ii) exactly one solution, or (iii) infinitely many solutions?
(d) If Aand Bare square matrices and AB =I, does BA =I? (Show this is true or provide
a counterexample.)
(e) Show that if Cis any m×nmatrix, then CTCis a symmetric matrix. (A complete answer
will include the definition of a symmetric matrix.)