# How do you find M^2 given M=((2,0), (0, 2))?

May 13, 2016

$\left(\begin{matrix}4 & 0 \\ 0 & 4\end{matrix}\right)$

#### Explanation:

$M = 2 I$ is a simple scalar multiple of the $2 \times 2$ identity matrix.

So ${M}^{2} = \left(2 I\right) \left(2 I\right) = 4 {I}^{2} = 4 I = \left(\begin{matrix}4 & 0 \\ 0 & 4\end{matrix}\right)$

Scalar multiplication commutes with matrix multiplication.

It is

${M}^{2} = \left(\begin{matrix}2 & 0 \\ 0 & 2\end{matrix}\right) \cdot \left(\begin{matrix}2 & 0 \\ 0 & 2\end{matrix}\right) = \left(\begin{matrix}2 \cdot 2 + 0 \cdot 0 & 2 \cdot 0 + 0 \cdot 2 \\ 0 \cdot 2 + 2 \cdot 0 & 0 \cdot 0 + 2 \cdot 2\end{matrix}\right) = \left(\begin{matrix}4 & 0 \\ 0 & 4\end{matrix}\right)$