<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] Verify that it is true for $n = 1$. This is the induction basis.
Then assume the result holds for $n$, so
$$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^n = \begin{pmatrix}a^n & na^{n-1}\\ 0 & a^n\end{pmatrix}$$
This is the induction hypothesis.
Finally, show (with the help of the induction hypothesis) that the result holds for $n + 1$, so
$$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^{n+1} = \begin{pmatrix}a^{n+1} & (n+1)a^n\\ 0 & a^{n+1}\end{pmatrix}$$
This can be easily proved:
\begin{align}
\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^{n+1} & = \begin{pmatrix}a & 1\\0 & a\end{pmatrix}\times\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^n\\
& = \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} & na^n + a^n\\0 & a^{n + 1}\end{pmatrix}\\
& = \begin{pmatrix}a^{n+1} & (n+1)a^n\\0 & a^{n + 1}\end{pmatrix}
\end{align}
as required.
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