# How do you multiply ((2, -1), (3, 4))^4?

Mar 22, 2018

$\left(\begin{matrix}- 107 & - 84 \\ 252 & 61\end{matrix}\right)$

#### Explanation:

Just like with ordinary numbers ${A}^{4} = A \times A \times A \times A$. This looks like it may take three matrix multiplications, but we can actually do this in 2.

$A = \left(\begin{matrix}2 & - 1 \\ 3 & 4\end{matrix}\right) \implies$
${A}^{2} = \left(\begin{matrix}2 & - 1 \\ 3 & 4\end{matrix}\right) \left(\begin{matrix}2 & - 1 \\ 3 & 4\end{matrix}\right) = \left(\begin{matrix}1 & - 6 \\ 18 & 13\end{matrix}\right)$
and thus
${A}^{4} = {A}^{2} \times {A}^{2} = \left(\begin{matrix}1 & - 6 \\ 18 & 13\end{matrix}\right) \left(\begin{matrix}1 & - 6 \\ 18 & 13\end{matrix}\right) = \left(\begin{matrix}- 107 & - 84 \\ 252 & 61\end{matrix}\right)$