I think i have this proof figured out but I just want a second opinion.
We have matrices $A=B$. $$A=\begin{pmatrix}a&1\\0&a\end{pmatrix}^n$$ $$B=\begin{pmatrix}a^n&n a^{n-1}\\0&a^n\end{pmatrix}$$
My base case I used $n=1$ and I was able to get them to equal. Next with my induction step I set $n=k$ and assumed that the base cases holds. So then I let $n=k+1$ and I got matrix $A$ to look nice. However when i did this process with matrix $B$ it did not equal matrix $A$.
So with this i continued to solve through and i have gotten them almost identical but i feel like i am missing a step.
Thank you!
Hint
If it is true for $n=k$ then: $$\begin{pmatrix}a&1\\0&a\end{pmatrix}^k=\begin{pmatrix}a^k&k a^{k-1}\\0&a^k\end{pmatrix}$$
so: $$\begin{pmatrix}a&1\\0&a\end{pmatrix}^{k+1}=\begin{pmatrix}a&1\\0&a\end{pmatrix}^{k} \cdot \begin{pmatrix}a&1\\0&a\end{pmatrix}=\begin{pmatrix}a^k&k a^{k-1}\\0&a^k\end{pmatrix} \cdot \begin{pmatrix}a&1\\0&a\end{pmatrix}$$ and by computing the last product you obtain: $$\begin{pmatrix}a^{k+1}&(k+1) a^{k+1-1}\\0&a^{k+1}\end{pmatrix}$$