18.06 Problem Set 6
Due Wednesday, 9 April 2008 at 4 pm in 2106.
Problem 1:
a) Do problem 2 from section 6.1 (pg. 283) in the book.
b) Do problem 9 from section 6.1 (pg. 284).
Problem 2:
Do problem 13 from section 6.1 (pg. 285) in the book.
Problem 3:
Consider the matrix
M
=
2
2
1
1

14

6

9

7

2

1

2

1
8
1
7
4
a) One eigenvector is
x
1
= (1
,
1
,
0
,

3). What is the corresponding eigenvalue?
b) Note that det(
M
) = 0. Use this information to ﬁnd another eigenvalue
λ
2

how do you know this must be an eigenvalue?
c) A third eigenvalue is
λ
3
=

1. Write down (but don’t solve) a linear system
that can be solved to ﬁnd
x
3
.
d) What is the fourth eigenvalue? (Hint: use the trace.)
Problem 4:
a) Do problem 8 in section 6.2 (pg. 299)
b) Do problem 18 in section 6.2 (pg. 300)
Problem 5:
Here’s an example of an invertible 3 by 3 matrix with only 2 diﬀerent
eigenvalues:
A
=
4 1

1
2 5

2
1 1
2
a) Find the eigenvalues of
A
.
b) Find 3 linearly independent eigenvectors of
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 Spring '08
 CHANILLO
 Linear Algebra, Matrices, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix

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