Cayley Hamilton Theorem

A matrix that satisfies its own characteristic polynomial

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$$\Delta (\lambda )=\det (\lambda \mathbb{I}-A)={\lambda }^n+{\alpha }_1{\lambda }^{n-1}+\cdots +{\alpha }_{n-1}A+{\alpha }_n$$ $$\downarrow$$ $$\Delta (A)=A^n+{\alpha }_1{A}^{n-1}+\cdots +{\alpha }_{n-1}A+{\alpha }_n\mathbb{I}=0$$