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?

For $n = 1$ we should have (according to the rule)
$$
\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^1 = \begin{pmatrix}a^1 & 1\cdot a^0\\0 & a^1\end{pmatrix}
$$
but of course
$$
 \begin{pmatrix}a^1 & 1\cdot a^0\\0 & a^1\end{pmatrix} = \begin{pmatrix}a & 1\\0 & a\end{pmatrix}
$$
so it's correct (this is called the base case and it's the starting point of the induction).
Now for the induction step. Assume that the rule is correct for all $n \leq k$ for some $k$ (this is called the induction hypothesis). We need to check, on the basis of this alone, that it also holds for $n = k+1$. We know that
$$
\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^{k+1} = \begin{pmatrix}a & 1\\0 & a\end{pmatrix}^k\cdot \begin{pmatrix}a & 1\\0 & a\end{pmatrix}
$$
Now, $\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^k$ we can rewrite (in red below), according to the induction hypothesis (remember, we assumed that the rule was true for all $n \leq k$). Therefore we have
$$
\color{red}{\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^k}\cdot \begin{pmatrix}a & 1\\0 & a\end{pmatrix} = \color{red}{\begin{pmatrix}a^k & ka^{k-1}\\0 & a^k\end{pmatrix}}\cdot \begin{pmatrix}a & 1\\0 & a\end{pmatrix}
$$
Now we just use matrix multiplication:
$$
\begin{pmatrix}a^k & ka^{k-1}\\0 & a^k\end{pmatrix}\cdot \begin{pmatrix}a & 1\\0 & a\end{pmatrix} = \begin{pmatrix}a^k \cdot a + ka^{k-1}\cdot 0 & a^k\cdot 1 + ka^{k-1}\cdot a\\ 0\cdot a + a^k \cdot 0& 0 \cdot 1 + a^k \cdot a\end{pmatrix}\\
= \begin{pmatrix}a^{k+1} & (k+1)a^k\\0 & a^{k+1}\end{pmatrix}
$$
which is exactly the rule for $n = k+1$, finishing the inductive step.
