Answer
$λ=-2$ is an eigenvalue
Work Step by Step
Given $$A=\begin{pmatrix}7&3\\3&-1 \end{pmatrix} $$
Since the characteristic equation
\begin{align*}
|A-\lambda I|&=0\\
\left| \begin{array}{rr} 7-\lambda&3\\3&-1-\lambda \end{array}\right|&=0\\
\left(7-λ\right)\left(-1-λ\right)-3\cdot \:3&=0\\
λ^2-6λ-16&=0\\
(-2)^2 - 6(-2) - 16 = \\
4 + 12 - 16= \\
16-16 = 0,
\end{align*}
Therefore $λ=-2$ satisfies the characteristic equation and $λ=-2$ is an eigenvalue of the given matrix