Question

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

Answer 1

$\text{LHS}:B+C=\left[\begin{array}{cc}-1& 6\\ 1& 10\end{array}\right]$

Now $A(B+C)=\left[\begin{array}{cc}3& 2\\ -1& 4\end{array}\right]\times \left[\begin{array}{cc}-1& 6\\ 1& 10\end{array}\right]=\left[\begin{array}{cc}-1& 38\\ 5& 34\end{array}\right]$

$\text{RHS}:AB=\left[\begin{array}{cc}3& 2\\ -1& 4\end{array}\right]\times \left[\begin{array}{cc}-2& 5\\ 6& 7\end{array}\right]=\left[\begin{array}{cc}6& 29\\ 26& 23\end{array}\right]$

$AC=\left[\begin{array}{cc}3& 2\\ -1& 4\end{array}\right]\left[\begin{array}{cc}1& 1\\ -5& 3\end{array}\right]=\left[\begin{array}{cc}-7& 9\\ -21& 11\end{array}\right]$

Now $AB+AC=\left[\begin{array}{cc}-1& 38\\ 5& 34\end{array}\right]$

Therefore, LHS $=$ RHS