Calculate matrix in negative power by using Cayley-Hamilton theorem

Question

I have found the characteristic polynomial of a 2x2 matrix $A$: $$λ^2-8λ+15=0$$ Through Cayley-Hamilton Theorem: $$A^2-8A+15I=0$$ We are asked to calculate $A^{-2}$ as a function of $A$ and $I$.

I would appreciate guidance.

Answer 1

So by simple rearrangement of your equation you have that $A^2 - 8A =-15I$ so

I=$\frac{-1}{15}(A^2-8A)=A(\frac{-1}{15}(A-8I))$

so $A^{-1}$ is $\frac{-1}{15}(A-8I)$

so $A^{-2} =\frac{-1}{15}(A-8I)\frac{-1}{15}(A-8I)=\frac{1}{15^2}(A(A-8I)-8I(A-8I))=\frac{1}{15^2}(A^2-8A-8A+64I)=\frac{1}{15^2}(A^2-16A+64I)$

but we know that $A^2$ is $8A -15I$ so $A^{-2}=\frac{1}{15^2}(A^2-16A+63I)=\frac{1}{15^2}(8A-15I-16A+64I)=\frac{1}{15^2}(-8A+49I)$

Hope this helps