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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

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