<s>[INST] Show that there isn't $A \in M_2(\mathbb R)$ such that $A^{2004}=\begin{pmatrix} -1 &0 \\ 0&-2 \end{pmatrix}$? What's the easiest and simplest way to show that there isn't $A \in M_2(\mathbb R)$ such that $A^{2004}=\begin{pmatrix}
-1 &0 \\ 
 0&-2 
\end{pmatrix}$?
Thank you.
 [/INST] Suppose $A^{2004}=\begin{pmatrix}
-1 &0 \\ 
 0&-2 
\end{pmatrix}.$
Then $B=A^{1002}$ is a real 2x2 matrix such that
$B^{2}=\begin{pmatrix}
-1 &0 \\ 
 0&-2 
\end{pmatrix}.$
We show that this is not possible. 
Let $\quad B=\begin{pmatrix}
a &b \\ 
 c&d 
\end{pmatrix}.$
Then
$B^{2}=\begin{pmatrix}
a^{2}+bc &ab+bd \\ 
 ac+cd&d^{2}+bc 
\end{pmatrix}=\begin{pmatrix}
-1 &0 \\ 
 0&-2 
\end{pmatrix}\quad \quad \quad  (1)$
Considering the (1,2) entry in equation (1) yields 
$$ab+bd=0.$$If $b \neq 0 $ then we must have
$a=-d$ 
but then  the diagonal entries of $B^{2}$ would be equal contradicting (1). If $b=0$ then by considering the (1,1) entry in equation (1) we see that $a^{2}=-1$, which is not possible for $a$ real.
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