MATH 323.502
Examination 1
February 23, 2015
NAME
SIGNATURE
“An Aggie does not lie, cheat, or steal or tolerate those who do.”
This exam consists of 5 problems, numbered 1–5. For partial credit you must present your
work clearly and understandably and justify your answers.
The use of calculators is not permitted on this exam.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 5 PROBLEMS ON
5 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1
2
3
4
5
Total
Points
Possible Credit
22
18
10
10
14
74
NAME
1.
MATH 323
Examination 1
Page 2
[22 points; (a) 4 pts.; (b)–(d) 6 pts. each] Consider the system of equations
4x + 8y = 4
3x + 6y − z = 0
3x + 6x − 2z = −3
(a) Write down the augmented matrix for this system of equations with the variables
ordered x, y, z.
(b) Carry out the row reduction of the augmented matrix in (a) and transform it into
reduced row echelon form. Show all of the steps you are taking.
(c) Determine the complete set of solutions of this system of equations.
(d) Is the coefficient matrix of this system invertible? Why or why not?
February 23, 2015
NAME
2.
MATH 323
Examination 1
Page 3
[18 points] For each statement below, write down whether it is true or false.
(a) A system of 6 linear equations in 3 variables must be inconsistent.
(b) A homogeneous system of 3 linear equations in 6 variables must have more than
one solution.
(c) If F1 and F2 are elementary matrices of type II, then F1 F2 = F2 F1 .
(d) For n × n matrices A and B, (AB)2 = A2 B 2 .
(e) For n × n matrices A and B, if AB is singular, then at least one of A and B must
be singular.
(f) The set W =
 
   
3
−1
0





a 2 + b 1 + 1 ∈ R3
0
0
1
a, b ∈ R is a subspace of R3 .
February 23, 2015
NAME
3.
MATH 323
Examination 1
Page 4
[10 points] Consider the matrix


1 0 −3
A = 0 1 0  .
0 2 1
Find two elementary matrices E1 and E2 of type III so that E2 E1 A = I.
4.
[10 points] Determine if the following sets of vectors are or are not vector spaces. In
each case, if the set is not a vector space, explain why not.
(a) V = the solution set of the equations x − 2y + 3z − 4w = 0 and x + 3z = 0 in R4 .
(b) W = the closed unit disk = {[ xy ] ∈ R2 | x2 + y 2 ≤ 1}.
February 23, 2015
NAME
5.
MATH 323
Examination 1
Page 5
[14 points] Let A and B be square matrices. We say that A is similar to B if there
exists an invertible matrix P so that A = P BP −1 .
(a) Prove that if A is similar to B, then B is similar to A.
(b) Prove that if A is similar to B, then det(A) = det(B).
February 23, 2015