# Show that (K1 +K2)A =K1A+K2A?

## If A= 0 1 2 |2 3 4| 4 5 6 and K1 =I, K2 = 2 show that (K1 +K2)A =K1A+K2A?

Apr 7, 2018

See below.

#### Explanation:

$\boldsymbol{A} = \left[\begin{matrix}0 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{matrix}\right]$

$K 1 = \left[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]$

$\left(K 1 + K 2\right) \boldsymbol{A} = \left[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right] \left[\begin{matrix}0 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{matrix}\right] + 2 \left[\begin{matrix}0 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{matrix}\right]$

=[(0,1,2),(2,3,4),(4,5,6)]+[(0,2,4),(4,6,8),(8,10,12)]=color(blue)([(0,3,6),(6,9,12),(12,15,18)]

$K 1 \boldsymbol{A} + K 2 \boldsymbol{A} = \left[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right] \left[\begin{matrix}0 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{matrix}\right] + 2 \left[\begin{matrix}0 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{matrix}\right]$

=[(0,1,2),(2,3,4),(4,5,6)]+[(0,2,4),(4,6,8),(8,10,12)]=color(blue)([(0,3,6),(6,9,12),(12,15,18)]