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The matrix$$ \mathbf{P} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} $$diagonalises$$ \mathbf{A} = \begin{pmatrix} 8 & -5 \\ 10 & -7 \end{pmatrix} $$with$$ \mathbf{P}^{-1} = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}\quad\text{and}\quad\mathbf{D} = \begin{pmatrix} 3 & 0 \\ 0 & -2 \end{pmatrix}. $$Calculate the matrix \(\mathbf{A}^4\).
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