Math 205 A,b Quiz 8 Worksheet - 2013

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MATH 205A,B - LINEAR ALGEBRA
WINTER 2013
QUIZ 8
NAME:
Section:(Circle one)
A(1 : 10)
B(2 : 40)
Show ALL your work CAREFULLY.
Let
3
0 0
A =
2 5 3
.
8
1 3
(a) The matrix A has 3 as one of its eigenvalues. Find the other eigenvalues of A.
The eigenvales of A are the solutions to the characteristic equation det(A
λI) = 0. It
follows that
3
λ
0
0
det(A
λI) = det
2
5
λ
3
8
1
3
λ
= (3
λ)[(5
λ)(3
λ)
(1)(3)]
2
= (3
λ)[λ
8λ + 12]
= (3
λ)((λ
6)(λ
2).
We now conclude that the other eigenvalues of A are 2 and 6.
(b) Find a basis for the eigenspace of A corresponding to the eigenvalue λ = 3.
When λ = 3, the matrix
0
0 0
8
1 0
1
1/8 0
1 1/8 0
1 1/8
0
1 0
1/6
A 3I =
2 2 3
2 2 3
2
2
3
0 9/4 3
0
1
4/3
0 1
4/3
8
1 0
0
0 0
0
0
0
0
0
0
0
0
0
0 0
0
Thus the eigenspace of A corresponding to λ = 3 is
1/6
1/6
2
4/3
4/3
x
: x
in
with basis
.
3
3
1
1
(c) Determine the rank of the matrix (A
3I). [Hint: what is the dimension of the eignespace as
in part (b)?] Justify your answer.
The rank of (A
3I) is the dimension of the column space of (A
3I). The eigenspace
in part (b), which is one dimensional, is the same as the null space of (A
3I). Since
(A
3I) is a 3
3 matrix, by the rank theorem, the rank of (A
3I) is equal to
3
dim Nul(A
3I) = 3
1 = 2.
Date: March 13, 2013.
1

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