Cayley-Hamilton Theorem

If $p(\lambda)$ is the characteristic polynomial of $A_{n\times n}$ , then $p(A)=O$ .

Every matrix satisfies its own characteristic equation.

Uses

Let $A = \begin{pmatrix}1 & 1 \\ 0 & 2\end{pmatrix}$ .

Solution Reveal Hide

Characteristic polynomial: $p(\lambda) = \lambda^2 - 3\lambda + 2$ .

By Cayley-Hamilton: $A^2 - 3A + 2I = O$ .

Computing $A^{-1}$ : multiply through by $A^{-1}$ :

A - 3I + 2A^{-1} = O \implies A^{-1} = \frac{3I - A}{2}

Computing $A^3$ : from $A^2 = 3A - 2I$ , multiply by $A$ :

A^3 = 3A^2 - 2A = 3(3A-2I) - 2A = 7A - 6I

Each higher power reduces to a linear combination of $A$ and $I$ .

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October 19, 2026 1 min read

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