<s>[INST] Matrix Power Formula 
Prove that for a fixed $a \in \mathbb{R}$ we have the matrix power formula for all $n \in \mathbb{Z}_+$:
  $$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^n = \begin{pmatrix}a^n & na^{n-1}\\0 & a^n\end{pmatrix}$$

How would we prove this? Right now we are doing work with proofs by induction... but how would I prove this?
 [/INST] By induction:
for $n = 1 $ it's true.
For $n +1 $ $$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^{n+1} = \begin{pmatrix}a & 1\\0 & a\end{pmatrix} \begin{pmatrix}a^n & na^{n-1}\\0 & a^n\end{pmatrix}  = \begin{pmatrix}a^{n+1} & a^n + na^n\\0 & a^{n+1} \end{pmatrix} = \begin{pmatrix}a^{n+1} & (n+1)a^n\\0 & a^{n+1} \end{pmatrix}$$ 
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