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APPM 2360: Midterm 2
June 30, 2013.
ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor’s
name, (3) your recitation section number and (4) a grading table. Text books, class notes,
and calculators are NOT permitted. A one page (2 sided, letter sized) crib sheet is allowed.
There are 5 questions and each is worth 20 points.
Problem 1:

(a)
(b)
(c)
(d)

1 2
−1
0
−3
Given the matrices A =  2 3  and B =
, which of the following
0 4
4
3 4
matrix operations are defined and which are not defined? (You do not need to give
reasons or evaluate those that are defined.)
(i) AB
(ii) A2
(iii) AT B
(iv) BB T
(v) A + B T
Give an example of three 2 × 2 matrices D, E and F , with E 6= F but DE = DF .
2 −3
−2 −5
and G3 =
.
Prove there does not exist a matrix G, with G2 =
6 −1
8 −7
(Hint: think determinants)
Are the functions f (t) = t2 − 2t + 1 and g(t) = 3t + 2 linearly independent?
Problem 2: Consider the following matrix and vector,
 


2
1 2 0 0 0
A =  2 4 1 1 1  , ~b =  3  .
λ
0 0 2 2 2
(a) (10 points) Col(A) is the span of the column vectors of matrix A. For what values
of λ is b in Col(A).
(b) (2 points) Find the dimension of Col(A).
(c) (5 points) Find the general solution to the linear homogeneous system Ax = 0 (i.e.,
describe the solution space).
(d) (3 points) Null(A) is the collection of all solutions to the homogeneous system Ax =
0. Construct a basis for Null(A) and find its dimension.
Problem 3: Let


0 −1 1
A = 3 3k 12 ,
1 0 3
where a and k are real numbers.
(a) Calculate det(A). (6 points)
 
 
x1
a
~x = x2  , ~b = 3 ,
x3
k
(b) After a few elementary-row
reduce [A|~b] to

1
 0
0
operations (which do not preserve the determinant), we

k
4
1
.
1 −1
−a
0 1 + k ak − k + 1
Find a and k such that A~x = ~b has
(i) one and only one solution, (3 points)
(ii) no solution, (3 points)
(iii) an infinite number of solutions. (3 points)
(c) Find x2 using Cramer’s rule when a = k = 0. (5 points)
Problem 4:
Let M22 denote the vector space consisting of all 2 × 2 matrices. Let V ∈ M22 be a 2 × 2
matrix such that V −1 exists. Define the subset W ⊆ M22 as
W = A ∈ M22 | A = V −1 DV
where
For example, one element of W may be
be any diagonal 2 × 2 matrix.
D can
2
0
23
0
V −1
V , while another may be V −1
V.
0 3
0 4
(a) (6 points) Is W a vector subspace? Justify your answer.
(b) (6 points) Prove that W is closed under matrix multiplication i.e. if A, B ∈ W,
then AB ∈ W. Hint: Show that the product of two (2 × 2) diagonal matrices is also
diagonal.
(c) (3 points) Show that AB = BA for any A, B ∈ W.
(d) (5 points) Define the magnitude of a matrix by the absolute value of the determinant,
|A| = |det(A)| ,
2 0
−1
where A ∈ W. Is the matrix A = V
V larger in magnitude than the
0 3
3 0
−1
matrix B = V
V ? Show your work.
0 1
Problem 5:
(a) Let A =
2 3
3 2
. Find the eigenvalues and corresponding eigenvectors. (10 points)


 
 
3
1
0





Let ~v1 = 4 , ~v2 = 1 , and ~v3 = 1  .
6
2
0
(b) Show that {~v1 , ~v2 , ~v3 } are linearly dependent. (5 points)
(c) What is the dimension of span ({~v1 , ~v2 , ~v3 })? (3 points)
(d) Is B = [~v1 |~v2 |~v3 ] invertible? (2 points)