APPM 3310-002: Matrix Methods — Final Exam — December 12, 2011
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.
Do problems 1, 2 and 3. Then, choose two of the three problems 4-6 on page 2. Indicate which
problem you are skipping by putting an X through that number on your grading sheet.
Please sign your bluebook under the Honor Code to indicate that you have neither given
nor received unauthorized assistance on this exam.
1. (30 points) Consider the set of equations 6x−8y+ 12z= 5 ,−3x+ 4y−6z=c, where cis
some real number.
(a) Write the system in matrix form Ax=b.
(b) Define the fundamental subspaces range and cokernel for an arbitrary matrix. Find these
spaces for A.
(c) Define the Fredholm compatibility conditions (Fredholm alternative) for a general
system Ax=b. Find a value of cthat satisfies these conditions for the system in (a).
(d) For this cvalue, find the general solution to (a). Write the solution as x=w+z, where
w∈corng(A) and z∈ker(A).
(e) Of the solutions in (d), which has the smallest Euclidean norm?
2. (30 points) State the fundamental theorem of linear algebraFor each property given below,
write down a matrix Awith that property or explain why no such matrix exists.
(a) rng(A) = span0
1, and corng(A) = span1
0.
(b) A=A−1.
(c) The vector
1
−1
2
is in the kernel of the A,
1
1
1
is in the corange of A, and det A= 1.
(d) Ais real and symmetric with eigenvalues 1 + iand 1 −i.
(e) Ahas an eigenvalue 2 with multiplicity two, but only one eigenvector.
3. (30 points) Let q(x, y, z) = x2+ 2xy + 3y2+ 4yz + 4z2be a quadratic form.
(a) Write q=xTKxfor a matrix K.
(b) Compute the LU decomposition of K.
(c) Define positive definite matrix. Is qa positive definite quadratic form? Explain.
(d) What do (a)-(c) allow you to conclude about the eigenvalues of K? (DO NOT COM-
PUTE the λ’s)
(e) λ1= 2 is an eigenvalue of K. Find its associated eigenvector.
(f) What is the sum of the other two eigenvalues, λ2+λ3? Note: DO NOT compute λ2and
λ3separately! Do not even think about wasting precious time doing this.
4. (20 points) Let S= span(x−1),(x−1)2.
(a) Define vector subspace.
