Why is $ \text{Rank}(A^{215}) = 3 $?

Question

I have a question. Why is $ \text{Rank}(A^{215}) = 3 $, where $$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 2 \end{bmatrix}? $$ How can I even calculate this? I’m sure that there’s an easy way to do this. I thought that maybe I would just calculate $ \text{Rank}(A) $, which is $ 4 $, and it would be the same after putting it in reduced row echelon form. However, the correct answer is $ 3 $. Why please?

Answer 1

Hint

The matrix $A$ is invertible if and only if $A^k$ is invertible for all $k\in\Bbb N$ if and only if $A$ is a full-rank matrix.