APPM 3310: Matrix Methods — Exam #1 — October 1, 2008
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Explain all of your answers. A correct answer with no supporting work may receive no
credit while an incorrect answer with some correct work may receive partial credit. No electronic
devices of any kind (e.g. cell phones, calculators, etc.) are permitted. Begin each problem on a new
page.
1. (30 points) For this problem, let A=
1 1 2 1
1 0 −1 3
2 3 7 0
(a) Find the LU decomposition of Awhere Uis in row echelon form.
(b) Determine the rank of Aand the dimensions of the four fundamental subspaces associated
with A.
(c) Find a basis for ker(A) and rng(A).
(d) Let b= [1 0 3]T. Use the LU decomposition from the previous part to find a solution to
the linear system Ax=b
2. (10 points)
(a) Describe three different ways you could tell whether a matrix is nonsingular.
(b) Now, suppose you know Ais n×nand Au=Avfor some u6=v. Is Asingular or
nonsingular? Explain.
3. (20 points) Let P(3) denote the vector space of all polynomials p(x) of degree less than or equal
to 3.
(a) Are p1(x) = x2−1, p2(x) = x2+ 1, p3(x) = 5, linearly independent elements of P(3)?
(b) What is the dimension of the subspace of P(3) spanned by p1,p2,p3?
4. (10 points) A square matrix is strictly upper triangular if all of the entries on or below the main
diagonal are zero.
(a) Show that that the set of all strictly upper triangular matrices is a subspace of Mn×n.
(b) Is the same true for the set of all special upper triangular matrices? Justify your answer.
5. (20 points) Answer the following True/False questions by either showing why the statement is
true in general or providing a counter example to show that it is false.
(a) True or False A set of vectors is linearly dependent if the zero vector belongs to their span.
(b) True or False The range of a matrix Ais the same as the span of its columns.
(c) True or False If Ais an m×nmatrix and rngA =Rmthen ker A =0.
6. (20 points) A matrix Jis skew-symmetric if JT=−J.
(a) If Jis skew-symmetric what do all of the entries on its diagonal have to be?
(b) Write down an example of a skew-symmetric matrix.
(c) Can you find a regular skew-symmetric matrix?
(d) Prove that if Jis a nonsingular skew-symmetric matrix then so is J−1.