Inverse via Cayley–Hamilton

HardLinear Algebra~6m

Use the Cayley–Hamilton theorem to express $A^{-1}$ as a linear combination of $A$ and $I$ for $A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$ , and report the top-left $(1,1)$ entry of $A^{-1}$ . (to 2 decimals)

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Write $p(\lambda)=\lambda^2-(\mathrm{tr})\lambda+\det$ , then $p(A)=0$ .
Multiply $A^2-5A+5I=0$ by $A^{-1}$ and solve for $A^{-1}$ .

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0.6 (± 0.01)

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Inverse via Cayley–Hamilton

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