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

• Compute the inverse of a matrix without row reduction.
• Express higher powers of a matrix in terms of lower powers.

Example

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

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

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

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

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

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