Finding Repeated Eigenvalues for $\begin{bmatrix} 3 & 0 & 0 \\ 0 & x & 2 \\ 0 & 2 & x \end{bmatrix}$

Question

I'm reviewing some linear algebra. I'm having trouble with this problem:

For what values of $x$ does the matrix
\begin{bmatrix}
3 & 0 & 0 \\
0 & x & 2 \\
0 & 2 & x
\end{bmatrix}
Have at least one repeated eigenvalue?

I'm not sure how to start this (aside from writing down the characteristic polynomial).

Edited: As per Shaun's direction.

Answer 1

\begin{equation} p(\lambda) =\det \begin{bmatrix} 3 - \lambda& 0 & 0 \\ 0 & x - \lambda& 2 \\ 0 & 2 & x - \lambda \end{bmatrix} = 0 \end{equation} So \begin{equation} (3-\lambda)( (x-\lambda)^2 - 4) = (\lambda - 3)(x - \lambda - 2)(x - \lambda + 2) = 0 \end{equation} So no matter what there is $\lambda_1 = 3$. Arguement goes on the other two eigenvalues $\lambda_2 = x - 2$ and $\lambda_3 = x + 2$. You've got the following cases:

• $\lambda_2 \neq \lambda_3$ no matter what.
• $\lambda_1 = \lambda_2$ gives $ x = 3 + 2 = 5$
• $\lambda_1 = \lambda_3$ gives $x = 3-2 = 1$

So two values of $x$ which are $1,5$.