$ X^{25} = A $ in $ M_{2}(\mathbb{R}) $

Question

Let $ A = \bigl(\begin{smallmatrix} 2& -1\\ 4& -2 \end{smallmatrix}\bigr) $

Find the number of solutions from $ M_{2}(\mathbb{R}) $ of the ecuation $ X^{25} = A $

Since $ A^2=O_{2} $ and $ det(A) = 0 $ $ \Rightarrow $ $ det(X^{25}) = 0 $ $ \Rightarrow $ $ det(X) = 0 $

Using Cayley Hamilton's formula we get that $ X^2=Tr(X) \cdot X $

$ X^{25} = A $ $ \Rightarrow $ $ X^{50} = A^2=O_{2} $

What to do next?

Answer 1

Since $X^2=tr(X) \cdot X$ you get $$0=X^{50}= (tr(X))^{49} \cdot X$$

Deduce from here that $tr(X)=0$ or $X=0$ and hence $$X^2=tr(X) \cdot X =0$$