Let $A = \left( \begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} \right)$ . Using elementary matrices, determine $A^n$ for any natural number $n$.

Question

Let $$A =\begin{pmatrix}1&0\\1&1\end{pmatrix}$$

Using what you know about elementary matrices, determine $A^n$ for any natural number $n$.

I am trying to solve the above question but do not understand what it's asking for. What does $A^n$ mean in terms of matrices?

Answer 1

$A^n$ means multiplying $ A$ $n$ times by itself, just as normal power with natural numbers. The matrix you have there is the elementary operation of adding the first row to second one. So:

$$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}.\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} a & b \\ {c+a} & {d+b} \end{bmatrix}$$

This means:

$$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}.\begin{bmatrix} 1 & 0 \\ n & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ n+1 & 1 \end{bmatrix}$$

Then: $$A^n=\begin{bmatrix} 1 & 0 \\ n & 1 \end{bmatrix}$$

Answer 2

Take your matrix $A$ and using what you know about matrix multiplication, multiply it by an arbitrary matrix, call it $B$. $$ B=\begin{pmatrix} a&b \\ c&d \end{pmatrix}\to A\cdot B=\begin{pmatrix}1&0\\1&1\end{pmatrix}\cdot \begin{pmatrix} a&b \\ c&d \end{pmatrix}= \begin{pmatrix} a & b \\ a+c & b+d \end{pmatrix} $$ You see that your matrix leaves the first row unchanged and each element in the bottom row is what it was before plus the entry above it. So you can see if you multiply $A$ by $A$, $n$ times the top row must remain $\begin{pmatrix} 1 & 0 \end{pmatrix}$. The first entry in the bottom row will be $1+1(n-1)=n$ and the remaining entry will be $1+0(n-1)=1$ (the $n-1$ is because following my example $B=A^{n-1}\to A^n=A\cdot A^{n-1}$).

So you have $$ A^n=\begin{pmatrix}1&0\\n&1\end{pmatrix} $$

Answer 3

The characteristic polynomial of $A$ is $$ p(\lambda)=\det(\lambda I-A)=(\lambda-1)^2. $$

Therefore $(A-I)^2=0$. It follows that $(A-I)^k=0$ for all $k\ge 2$ and, hence, \begin{align} A^n &= (I+(A-I))^n \\ &=I+n(A-I) \\ &=\begin{pmatrix}1 & 0 \\ n & 1\end{pmatrix}. \end{align}