If $\mathrm{A}=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$ and ${\mathrm{A}}^2-\mathrm{\lambda A}-{\mathrm{I}}_2=\mathrm{O} ,$ then $\lambda$ is equal to 

$$\begin{gathered} \mathrm{A}=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right] \\ {A}^2=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]=\left[\begin{array}{cc}5& 8\\ 8& 13\end{array}\right] \\ \Rightarrow {\mathrm{A}}^2-\mathrm{\lambda A}-{\mathrm{I}}_2=\mathrm{O} \Rightarrow {\mathrm{A}}^2-{\mathrm{I}}_2=\mathrm{\lambda A}\Rightarrow \mathrm{\lambda A}=\left[\begin{array}{cc}4& 8\\ 8& 12\end{array}\right]=4\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right] \\ \Rightarrow \mathrm{\lambda }=4 \end{gathered}$$