If `A=[[3,-4],[1,-1]]`, then prove that `A^n=[[1+2n,-4n],[n,1-2n]]`, where n is any positive integer.

Explanation

To prove that \( A^n = \begin{bmatrix} 1 + 2n & -4n \\ n & 1 - 2n \end{bmatrix} \) for the matrix \( A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \), we will compute the first few powers of \( A \) and look for a pattern.

### Step 1: Calculate \( A^1 \)

Given:
\[ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \]
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