- #1
DanielFaraday
- 87
- 0
Homework Statement
Find the eigenvalues of the following matrix:
[tex]
\left(
\begin{array}{ccc}
1 & 0 & -3 \\
1 & 2 & 1 \\
-3 & 0 & 1
\end{array}
\right)
[/tex]
Homework Equations
The Attempt at a Solution
I think I'm forgetting a basic algebra rule or something. I know there are supposed to be 3 eigenvalues, but I am only getting 2.
[tex]
0=\det (\lambda I-A)=\left|
\begin{array}{ccc}
\lambda -1 & 0 & 3 \\
-1 & \lambda -2 & -1 \\
3 & 0 & \lambda -1
\end{array}
\right|
[/tex]
[tex]
=\frac{(\lambda -1)(\lambda -2)(\lambda -1)-3(3)(\lambda -2)}{(\lambda -2)}
[/tex]
[tex]
=(\lambda -1)(\lambda -1)-9
[/tex]
[tex]
=(\lambda -1)^2-9
[/tex]
[tex]
(\lambda -1)^2=9
[/tex]
[tex]
\lambda =\pm 3+1
[/tex]