Let $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$  Find $\mathbf{A}^{20} - 2 \mathbf{A}^{19}.$
Question:
Let $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$  Find $\mathbf{A}^{20} - 2 \mathbf{A}^{19}.$
Answer:
Let's think step by step.

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\begin{pmatrix} 0 & 3 \\ 0 & -1 \end{pmatrix}
Let $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$  Find $\mathbf{A}^{20} - 2 \mathbf{A}^{19}.$
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