Cayley–Hamilton para matrizes \(2\times2\)

Para \(A = \left(\begin{smallmatrix}a&b\\c&d \end{smallmatrix}\right)\), \(p_A(\lambda) = \lambda^2-(a+d)\lambda+(ad-bc)\), e uma conta direta dá \[A^2-(a+d)A+(ad-bc)I = \begin{pmatrix}a^2+bc&ab+bd\\ac+cd&bc+d^2\end{pmatrix} -\begin{pmatrix}a^2+ad&ab+bd\\ac+cd&ad+d^2\end{pmatrix}+\begin{pmatrix}ad-bc&0\\0&ad-bc \end{pmatrix} = 0.\]