Calculate the product $A^{10}v$ where $A$ is a $2 \times 2$ matrix and $v$ is a vector $[4 \; 4]^T$

Question

Following on from the title, can someone suggest how to proceed with this one.

$$A=\begin{bmatrix}1&1\\4&1\end{bmatrix}$$

and

$$v = [4 \; 4]^T?$$

Answer 1

You can

• calculate $Av$, and then apply $A$ to the result nine times;

• or, since the eigenvalues of $A$ are distinct, $A$ is diagonalizable: there exist $P$ invertible and $D$ diagonal with $A=PDP^{-1}$. Then $A^{10}=PD^{10}P^{-1}$, where $D^{10}$ is very easy since it's diagonal.