marix proof by induction

Question

Given the matrix $A$

\begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}

use proof by induction to show that $A^n$ for $n=1,2..$. is

\begin{pmatrix} 2^n & 2^n - 1 \\ 0 & 1 \end{pmatrix}

so assuming $n=k$ and trying to show for $n=k+1$ .I managed to get the equation below, but im not sure how to show that $LS = RS$ on this problem?(I might have done something wrong), any tips or solutions? thanks!

$A^{k+1} =\begin{pmatrix} 2^{k+1} & 2^{k+1} - 1 \\ 0 & 1 \end{pmatrix}$

Answer 1

hint:$$A^{n+1} = A^n\cdot A = \begin{pmatrix} 2^n & 2^n - 1 \\ 0 & 1 \end{pmatrix}\cdot \begin{pmatrix} 2 & 1\\ 0 & 1\end{pmatrix}=....$$