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\section*{Άσκηση 4}

Αν 
$$
A = \begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix},
$$
υπολογίστε τον πίνακα \(A^n\).

\subsection*{Λύση}

Έχουμε
$$
A^2 \;= A\cdot A
=\begin{pmatrix}0&1\\-1&0\end{pmatrix}
 \begin{pmatrix}0&1\\-1&0\end{pmatrix}
 =\begin{pmatrix}-1&0\\0&-1\end{pmatrix}
 =-I_2.
 $$
 $$
 A^3 = A^2\cdot A = -I_2\cdot A = -A,
 \qquad
 A^4 = A^2\cdot A^2 = (-I_2)\cdot(-I_2)=I_2.
 $$
 Άρα
 \[
 A^n = 
 \begin{cases}
 I_2,   & n = 4k,\\
 A,     & n = 4k + 1,\\
 -\,I_2,& n = 4k + 2,\\
 -\,A,  & n = 4k + 3,
 \end{cases}
 \quad k\in\mathbb{N}\\
 
 \]

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