$A-\lambda I=\left[\begin{array}{ll}1 & 3 \\ 2 & 2\end{array}\right]-\lambda\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}1-\lambda & 3 \\ 2 & 2-\lambda\end{array}\right]$
Since $A-\lambda I$ is a singular matrix,
$\left|\begin{array}{cc}1-\lambda & 3 \\ 2 & 2-\lambda\end{array}\right|=0$
$\begin{aligned} & (1-\lambda)(2-\lambda)-6=0 \\ & 2-3 \lambda+\lambda^2-6=0 \\ & \lambda^2-3 \lambda-4=0\end{aligned}$
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