MATH 323.501
Examination 1
October 1, 2013
NAME
SIGNATURE
“An Aggie does not lie, cheat, or steal or tolerate those who do.”
This exam consists of 7 problems, numbered 1–7. 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 7 PROBLEMS ON
6 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1
2
3
4
5
6
7
Total
Points
Possible Credit
16
10
15
10
11
10
15
87
NAME
1.
MATH 323
Examination 1
Page 2
[16 points; (a) 4 pts.; (b) & (c) 6 pts. each] Consider the system of equations
3x − 3z = 9
4x − 2y − 8z + w = 21
2y + 4z = −4
(a) Write down the augmented matrix for this system of equations with the variables
ordered x, y, z, w.
(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.
October 1, 2013
NAME
2.
MATH 323
Examination 1
Page 3
[10 points] Consider the matrix


0 1 1
A = −1 1 0 .
2 0 1
Find the inverse matrix of A, and show all of the steps that you take.
October 1, 2013
NAME
3.
MATH 323
Examination 1
Page 4
[15 points] For each statement below, write down whether it is true or false.
(a) A system of 4 linear equations in 5 variables must have more than one solution.
(b) For invertible n × n matrices A and B, (AB)−1 = A−1 B −1 .
(c) For an n×n matrix A of determinant 5, if B is the matrix obtained by multiplying
one of the rows of A by 3 and then interchanging two other rows, then the determinant
of B must be 15.
(d) For an elementary matrix E, the matrix E T must be an elementary matrix.
(e) For a positive integer d, the set of all polynomials of degree d with real coefficients
is a vector space.
4.
[10 points] Find all values of x such that the matrix
6
x+3
A=
x−2
x
is singular. Explain.
October 1, 2013
NAME
5.
MATH 323
Examination 1
Page 5
[11 points; (a) 3 pts., (b) 8 pts.] Let B be the matrix


1 −3 0 2
B = 0 0 1 −1 ,
0 0 0 0
and let N (B) be its nullspace.
(a) N (B) is a subspace of Rn for what value of n? Explain.
(b) Find two vectors a, b so that N (B) = Span(a, b).
6.
[10 points] Consider the matrices


3 2 1
A = 1 0 2  ,
1 1 −2


1 0 5
B = 4 2 3  .
1 1 −2
Find elementary matrices E1 and E2 so that B = E1 E2 A. (Hint: You can take both
E1 and E2 to be of type III.)
October 1, 2013
NAME
7.
MATH 323
Examination 1
Page 6
[15 points] Let V be a vector space, and suppose that W1 and W2 are both subspaces
of V . Let
W = {w1 + 3w2 | w1 ∈ W1 , w2 ∈ W2 }.
Prove that W is a subspace of V .
October 1, 2013