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Consider the matrix \(\mathbf{A} = \begin{pmatrix} 5 & -2 \\ 4 & -1 \end{pmatrix}\).
  1. Find the eigenvalues \(\lambda_1\) and \(\lambda_2\) of matrix \(\mathbf{A}\).
  2. Find the corresponding eigenvectors \(\mathbf{x}_1\) and \(\mathbf{x}_2\).
  3. Determine the matrices \(\mathbf{D}\), \(\mathbf{P}\), and \(\mathbf{P}^{-1}\) such that \(\mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\).
  4. Hence, calculate the matrix \(\mathbf{A}^6\).

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