Math 317: Linear Algebra
Exam 3
Fall 2015
Name:
*No notes or electronic devices. You must show all your work to receive full credit. When
justifying your answers, use only those techniques that we learned in class up to Section
5.1. I am thinking of the number thirty seven. Unless otherwise stated, you do not have
to simplify your answers. Good luck!
1. Suppose that A is a 4 × 4 matrix with rows a, b, c and d such that det A = 5. That
is,
 
a
b

A=
c
d
(a) (10) Calculate det B if


3a + b + d


b

B=
 c+d 
d
(b) (5) Calculate det A−1 or explain why it does not exist.
(c) (5) Calculate det AB (if possible). If it is not possible, explain why.
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Math 317: Linear Algebra
Exam 3
Fall 2015
2. Let V = Span ((1, 0, 1, 0), (0, 1, 4, 2)) ⊂ R4 .
(a) (5) Find an orthogonal basis for V .
(b) (5) Find a basis for V ⊥ .
(c) (5) Find the projection of (1, 1, 1, 1) on V ⊥ . Hint: Use the orthogonal basis
you obtained in (a) to your advantage.
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Math 317: Linear Algebra
Exam 3
Fall 2015
3. Let A be an n × n matrix and let Rn×n denote the space of all n × n matrices.
(a) (5) Define the trace of a matrix A.
(b) (10) Let S = {A ∈ Rn×n | BA is symmetric}. Prove that S is a subspace of
Rn×n .
3
Math 317: Linear Algebra
Exam 3
Fall 2015
4. (a) (5) Suppose that A is a n × n matrix. Prove that det(cA) = cn det A for any
nonzero c ∈ R.
(b) (5) Suppose that A is a skew-symmetric matrix. (Recall that this means that
A = −AT .) Prove that A is a singular matrix if n is odd. (Hint: Recall that
det(A) = det(AT ).)
4
Math 317: Linear Algebra
Exam 3
Fall 2015
 
 
1
1
3



5. (25, 5 each) Let V ⊂ R be the plane spanned by v1 = 0 and v2 = 1.
2
0
(a) Find a vector v3 so that V ⊥ = span(v3 ). Note that B = {v1 , v2 , v3 } is a basis
for R3 .
(b) Let T1 : R3 → R3 be the linear transformation which projects x ∈ R3 onto
V ⊥ . Find [T1 ]B , the matrix of T with respect to the basis B.
(c) Let T2 : R3 → R3 be the linear transformation which reflects x ∈ R3 over V .
Find [T2 ]B , the matrix of T2 with respect to the basis B.
(d) Use the change of basis formula to compute [T2 ]stand . That is, compute the
standard matrix of T2 . You do not need to carry out the matrix multiplication/inversion.
(e) Calculate det [T2 ]stand .
5
Math 317: Linear Algebra
Exam 3
Fall 2015
6. (5 points each) Determine if the following statements are true or false. If the statement is true, give a brief justification. If the statement is false, provide a counterexample or explain why the statement is not true.
(a) If A = QR where Q is an orthogonal matrix and R is an upper triangular matrix
with positive diagonal entries, then the least squares solution x̄ associated with
Ax = b can be written as x̄ = R−1 QT b.
(b) There is a nonsingular matrix P such that
1 2
3 1
P =P
.
1 0
1 0
1 2
1 2
(c) det
= det
.
3 4
4 6
6
Math 317: Linear Algebra
Exam 3
Fall 2015
7. (5 points) (Bonus): I am thinking of a number between 1-100. What is it?
7