Matrix to the power 100 by Cayley Hamilton

Question

I have the matrix

$$A = \begin{pmatrix} 1 & 1 & −1 \\ −1 & −1 & 0 \\ 0 & −1& 0\end{pmatrix}$$

with characteristic polynomial $p(x)=-x^3 + x - 2$ and I have to find $A^{100}$ with the Cayley-Hamilton theorem. Can anybody help me?

Answer 1

$$A=\begin{pmatrix}1&1&\!-1\\\!-1&\!-1&0\\0&\!-1&0\end{pmatrix}\implies \det(xI-A)=\begin{vmatrix}x-1&\!-1&1\\1&x+1&0\\0&1&x\end{vmatrix}=$$$${}$$

$$=x(x^2-1)+x+1=x^3+1\implies\;\text{by C.H.,}\;A^3=-I\implies $$

$$A^{100}=(A^3)^{33}A=-A$$