Suppose that the matrix $A$ is as follows: $A=\begin{bmatrix} -3&4\\ 2&3\end{bmatrix}$
We need to prove that $A^{2n + 1}=A$.
The way I tackled this problem is as follows:
- If $A^{2n + 1} =A$, then $A^{2n}$ must be the same as the identity matrix $I$.
- Thus $(A^2)^n$ must be the same as $I$.
- By calculation $A^2=17I$.
- Thus the statement can't be proved.
I'm not sure if I'm correct. I believe we have to make use of Cayley Hamilton's Theorem and diagonalization to solve the problem but I can't seem to wrap my head around it. Any help will be appreciated.