For 1x1 let A = $\left | 2 \right |$
its characteristic equation will be A= $\left | 2-\lambda \right |$=0
Re arranging in above format
f(x) = 2 - 1*$\lambda$
det = $(-1)^{1}\frac{2}{-1}$= 2
2x2 matrix A =$\begin{vmatrix}1\hspace{1cm}2&\\3\hspace{1cm}4&\end{vmatrix}$
characteristic equation will be$\begin{vmatrix} 1-\lambda\hspace{1cm} 2 & \\ 3\hspace{1cm} 4-\lambda & \end{vmatrix}$=0
(1-$\lambda$ )(4-$\lambda$ )-6
f(x)=$\lambda ^{2}-5\lambda -2=0$
det(A) = $(-1)^{2}\frac{-2}{1}$= -2