Math 2270
Quiz 12
1. Find the eigenvalues of the matrix
A=
6
2
3
7
.
We must have
0 = det(A − λI) = det
6−λ
3
2
7−λ
⇒ λ2 − 13λ +
= (6 − λ)(7 − λ) − 6 = 42 − 6λ − 7λ + λ2 − 6 = λ2 − 13λ + 36.
169
169
= −36 +
⇒
4
4
2
25
13
13 5
=
λ−
⇒ λ=
± = 9 or 4.
2
4
2
2
2. Find a basis for each eigenspace of the matrix in problem 1.
From problem 1, we know that the eigenvalues are λ1 = 9 and λ2 = 4. To find the corresponding eigenvectors,
we row reduce:
6−9
3
−3 3
x=r
1 −1
1
=
∼
⇒
⇒ ~v1 =
,
2
7−9
2 −2
y=r
0 0
1
6−4
2
3
7−4
=
2 3
2 3
∼
1
0
3
2
0
Thus, {~v1 } is a basis for Eλ1 and {~v2 } is a basis for Eλ2 .
⇒
x = − 23 r
y=r
⇒ ~v2 =
−3
2
.