If $A = [2\,\,\,\,\,\,\,1],$ $B = \left[ {\begin{array}{llllllllllllllllllll}5&3&4\\8&7&6\end{array}} \right]$ and $C = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&2&1\\1&0&2\end{array}} \right]$, then verify that
$A(B + C) = (AB + AC)$.

We have to verify that, $A(B + C) = AB + AC$

We have, $A = \left[ {\begin{array}{llllllllllllllllllll}2&1\end{array}} \right],B$

$= \left[ {\begin{array}{llllllllllllllllllll}5&3&4\\8&7&6\end{array}} \right]$

and $C = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&2&1\\1&0&2\end{array}} \right]$

$\therefore$ $A(B + C) = [2\,\,\,\,\,\,1]\left[ {\begin{array}{llllllllllllllllllll}{5 - 1}&{3 + 2}&{4 + 1}\\{8 + 1}&{7 + 0}&{6 + 2}\end{array}} \right]$

$= [2\,\,\,\,\,1]\left[ {\begin{array}{llllllllllllllllllll}4&5&5\\9&7&8\end{array}} \right]$

$= [8 + 9\,\,\,\,\,10 + 7\,\,\,\,\,\,\,10 + 8]$
$= [17\,\,\,\,17\,\,\,\,\,18]$

…….(i)
$AB = [21]\left[ {\begin{array}{llllllllllllllllllll}5&3&4\\8&7&6\end{array}} \right]$

$= [\begin{array}{llllllllllllllllllll}{10 + 8}&{6 + 7}&{8 + 6]}\end{array} = \begin{array}{llllllllllllllllllll}{[18}&{13\,\,\,\,}\end{array}\,\,\,\,14]$

and $AC = [2\,\,\,\,\,1]\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&2&1\\1&0&2\end{array}} \right]$

$= \left[ {\begin{array}{llllllllllllllllllll}{ - 2 + 1}&{4 + 0}&{2 + 2}\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}{ - 1}&4&4\end{array}} \right]$

$\therefore$ $AB + AC = \left[ {\begin{array}{llllllllllllllllllll}{18}&{13}&{14}\end{array}} \right] + \left[ {\begin{array}{llllllllllllllllllll}{ - 1}&4&4\end{array}} \right]$

$= \left[ {\begin{array}{llllllllllllllllllll}{17}&{17}&{18}\end{array}} \right]$

…….(ii)
$\therefore$ $A(B + C) = (AB + AC)$

[using Eqs. (i) and (ii)]