The Cayley-Hamilton theorem

Characteristic polynomial. $\chi_A(X) = \det(A - XI) = (2-X)^2 - 1 = X^2 - 4X + 3$.

Compute $\chi_A(A)$. We have
$$A^2 = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}^2 = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix}.$$
Then
$$\chi_A(A) = A^2 - 4A + 3I = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix} - \begin{pmatrix} 8 & 4 \\ 4 & 8 \end{pmatrix} + \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}.$$

Conclusion. $\chi_A(A) = 0$, confirming Cayley-Hamilton.

Application. This gives $A^2 = 4A - 3I$, which allows computing any power of $A$ by recurrence.