Question

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],B=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$ and $C=\left[\begin{array}{cc}k& l\\ m& n\end{array}\right]$ then find $AB+AC$

Answer 1

$AB=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$

$=\left[\begin{array}{cc}ap+br& aq+bs\\ cp+dr& cq+ds\end{array}\right]$

$AC=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}k& l\\ m& n\end{array}\right]$

$=\left[\begin{array}{cc}ak+bm& al+bn\\ ck+dm& cl+dn\end{array}\right]$

$AB+AC=\left[\begin{array}{cc}ap+br& aq+bs\\ cp+dr& cq+ds\end{array}\right]+\left[\begin{array}{cc}ak+bm& al+bn\\ ck+dm& cl+dn\end{array}\right]$

$=\left[\begin{array}{cc}ap+br+ak+bm& aq+bs+al+bn\\ cp+dr+ck+dm& cq+ds+cl+dn\end{array}\right]$