[Math] Find $A^{1000}$ by using Cayley-Hamilton Theorem

linear algebramatrix equations

I get stuck at the following question:

Consider the matrix
$$A=\begin{bmatrix}
0 & 2 & 0 \\
1 & 1 & -1 \\
-1 & 1 & 1\\
\end{bmatrix}$$

Find $A^{1000}$ by using the Cayley-Hamilton theorem.

I find the characteristic polynomial by $P(A) = -A^{3} + 2A^2 = 0$ (by Cayley-Hamilton) but I don't see how to find $A^{1000}$ by this characteristic polynomial.

Best Answer

Your formula tells you, after you multiply through by $A^{997}$, that $$A^{1000}=2A^{999}.$$ Similarly, $$2A^{999}=4A^{998}.$$

This process can be repeated to find $A^{1000}$ in terms of $A^2$, which you can then compute.

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