The Cayley-Hamilton theorem

Theorem. Every square matrix satisfies its own characteristic polynomial: $\chi_A(A)=0$.

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

Verification.
$$A^2 = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix}.$$
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. This gives $A^2=4A-3I$, allowing recursive computation of $A^k$.