<s>[INST] Find the minimum positive integer $n$ and the matrix $A^{2020}$ Let $A=a\begin{pmatrix} 1 & -1\\
1 & 1 \end{pmatrix}$ $(a>0)$ and $I=\begin{pmatrix} 1 & 0\\
0 & 1 \end{pmatrix}$ satisfy $A^4+I=\begin{pmatrix} 0 & 0\\
0 & 0 \end{pmatrix}$

*

*Find $a$ For this I can do. I saw $a=\frac{1}{\sqrt{2}}$

*Find the minimum positive integer $n$ such that $A^n\begin{pmatrix} 0\\ 1 \end{pmatrix}= \begin{pmatrix} 1\\ 0 \end{pmatrix}$

*Find $A^{2020}$

So please help to tell me about this 2 and 3. Give me some hints or ideas!!
 [/INST] For 2: $A^{2}\left(\begin{array}{c}
0\\
1
\end{array}\right)=\left(\begin{array}{c}
-1\\
0
\end{array}\right)$, so$$A^6\left(\begin{array}{c}
0\\
1
\end{array}\right)=-A^2\left(\begin{array}{c}
0\\
1
\end{array}\right)=\left(\begin{array}{c}
1\\
0
\end{array}\right).$$Hint for 3: $2020$ is an odd multiple of $4$.
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