Solution
= Solution
Using the convention of the question, the <characteristic polynomial> and characteristic equation are
$$
\chi_A(z)=\det(A-zI),
\qquad
\chi_A(z)=0.
$$
For
$$
A=\begin{pmatrix}a&b\\c&d\end{pmatrix},
$$
one has
$$
\chi_A(z)=z^2-(a+d)z+(ad-bc).
$$
Direct multiplication gives
$$
A^2-(a+d)A+(ad-bc)I=0,
$$
entry by entry. Thus $\chi_A(A)=0$, verifying the <Cayley-Hamilton theorem> for $2\times2$ matrices.