<s>[INST] How to find eigenvalues of matrix $\begin{bmatrix} 3& a+1\\a+1&3 \end{bmatrix}$ I want to find the eigenvalues of the following matrix
$\begin{bmatrix} 3& a+1\\a+1&3 \end{bmatrix}$ expressed in $a$ using $\begin{bmatrix} \lambda - 3& a+1\\a+1&\lambda-3 \end{bmatrix}$.
But the $a$-term makes it difficult for me to find it. I hope someone can show me how to do this. Thanks in advance !
EDIT:
Determinant = $4a^2+8a+4$
Using the abc rule I get
$(6\overset{+}{-} \sqrt{4a^2+8a+4})/2$
 [/INST] \begin{align}
\left| \begin{matrix} \lambda -3 & a+1 \\
a+1 &\lambda-3 \end{matrix} \right| & = (\lambda-3)(\lambda-3) - (a+1)(a+1) \\
&= \lambda^2-6\lambda+9-(a^2+2a+1) \\
&= \lambda^2-6\lambda+9-a^2-2a-1 \\
&= \lambda^2-6\lambda-a^2-2a+8 \\
&= \lambda^2-6\lambda-(a^2+2a-8)\\
&= \lambda^2-6\lambda-(a+4)(a-2) \\
&= (\lambda -a-4)(\lambda +a-2) =0\\
\end{align}
Thus $\lambda = a+4$ and $\lambda = -a+2$.
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