If matrix a satisfies the equation `A^(2)=A^(-1)`, then prove that `A^(2^(n))=A^(2^((n-2))), n in N`.

Explanation

To prove that if matrix \( A \) satisfies the equation \( A^2 = A^{-1} \), then \( A^{2^n} = A^{2^{n-2}} \) for \( n \in \mathbb{N} \), we can follow these steps:

### Step 1: Rewrite \( A^{2^n} \)
We start by rewriting \( A^{2^n} \) in terms of \( A^{2^{n-1}} \):
\[ A^{2^n} = A^{2^{n-1} \cdot 2} = A^{2^{n-1}} \cdot A^{2^{n-1}} \]
**Hint:** Use the property of exponents that states \( a^{m \cdot n} = (a^m)^n \).
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