Question

$A=\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right),B=\left(\begin{array}{cc}-2& 5\\ 6& 7\end{array}\right)$ and $C=\left(\begin{array}{cc}1& 1\\ -5& 3\end{array}\right)$, then verify that $A(B+C)=AB+AC$

Answer 1

Given, $A=\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right)$ $B=\left(\begin{array}{cc}-2& 5\\ 6& 7\end{array}\right)$ $C=\left(\begin{array}{cc}1& 1\\ -5& 3\end{array}\right)$
Thus, $B+C=\left(\begin{array}{cc}-2& 5\\ 6& 7\end{array}\right)+\left(\begin{array}{cc}1& 1\\ -5& 3\end{array}\right)=\left(\begin{array}{cc}-1& 6\\ 1& 10\end{array}\right)$
$\Rightarrow A(B+C)=\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right)\left(\begin{array}{cc}-1& 6\\ 1& 10\end{array}\right)$
$=\left(\begin{array}{cc}-3+2& 18+20\\ 1+4& -6+40\end{array}\right)=\left(\begin{array}{cc}-1& 38\\ 5& 34\end{array}\right)$ .......(1)
$\Rightarrow AB=\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right)\left(\begin{array}{cc}-2& 5\\ 6& 7\end{array}\right)$ and $AC=\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right)\left(\begin{array}{cc}1& 1\\ -5& 3\end{array}\right)$
$\Rightarrow AB=\left(\begin{array}{cc}-6+12& 15+14\\ 2+24& -5+28\end{array}\right)$ and $AC=\left(\begin{array}{cc}3-10& 3+6\\ -1-20& -1+12\end{array}\right)$
$\Rightarrow AB=\left(\begin{array}{cc}6& 29\\ 26& 23\end{array}\right)$ and $AC=\left(\begin{array}{cc}-7& 9\\ -21& 11\end{array}\right)$
$\Rightarrow AB+AC=\left(\begin{array}{cc}6& 29\\ 26& 23\end{array}\right)+\left(\begin{array}{cc}-7& 9\\ -21& 11\end{array}\right)$
$=\left(\begin{array}{cc}-1& 38\\ 5& 34\end{array}\right)$ .........(2)
From (1) and (2), we have
$A(B+C)=AB+AC$