Proof by induction: Matrices

Question

Given the matrix $A=\begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}$, I want to prove that $A^k = \begin{pmatrix}1 & 2k \\ 0 & 1 \end{pmatrix}$ (=induction hypothesis). Since I struggled a bit with induction in the past, I want to know if I did this correctly:

$A^1 $ is clear.

$A^{k+1} = \begin{pmatrix} 1 & 2(k+1)\ \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 1 & 2k \\ 0 & 1\end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}$ with induction hypothesis: $ A^{k+1} = A^k \cdot \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}$ which is basically what we are looking for.

Answer 1

You must assume truth for $\;k\;$ , i.e.:

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

and under this assumption prove for $\;k+1\;$ , but

$$A^{k+1}=\color{red}{A^k}A\stackrel{\color{red}{\text{Ind. Hypothesis}}}=\color{red}{\begin{pmatrix}1&2k\\0&1\end{pmatrix}}\begin{pmatrix}1&2\\0&1\end{pmatrix}$$

Now carry on the easy matrix product on the right and verify you get what you need.

Answer 2

Hint $\ $ It is the special case $\ a_j = 2\ $ of the following

$$ \left[\begin{matrix} 1 & a_{k+1} \\ 0 & 1 \end{matrix}\right] \left[\begin{matrix} 0 & a_1+\cdots+a_{k} \\ 0 & 1 \end{matrix}\right] \,=\, \left[\begin{matrix} 1 & a_1+\cdots+a_{k+1} \\ 0 & 1 \end{matrix}\right]\qquad\ $$

$$\Rightarrow\quad {\Large \prod_{j=1}^k} \left[\begin{matrix} 1 & a_{j} \\ 0 & 1 \end{matrix}\right]\,=\, \left[\begin{matrix} 1 & \sum_{j=1}^k a_{j} \\ 0 & 1 \end{matrix}\right] $$