Question

Use $A=\left[\begin{array}{cc}1& -2\\ 4& 3\end{array}\right]$,$B=\left[\begin{array}{cc}-5& 2\\ 4& 3\\ & \end{array}\right]$, and$C=\left[\begin{array}{cc}5& 1\\ 2& -4\\ & \end{array}\right]$and scalar $c=3$to determine whether the equation is true for the given matrices.

$C\left(A+B\right)=AC+BC$

Short answer

The provided equation is false.

Step-by-step solution

Step 1. Determine the value of CA+B

Substitute the provided matrices into $C\left(A+B\right)$.

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

First perform the addition after it multiply the resulting matrix with C.

$\begin{array}{c}C\left(A+B\right)=\left[\begin{array}{cc}5& 1\\ 2& -4\end{array}\right]\cdot \left[\begin{array}{cc}-4& 0\\ 8& 6\end{array}\right]\\ =\left[\begin{array}{cc}1& 1\\ 10& 2\end{array}\right]\end{array}$

Step 2. Determine the value of AC+BC

Substitute the provided matrices into $AC+BC$.

$AC+BC=\left[\begin{array}{cc}1& -2\\ 4& 3\\ & \end{array}\right]·\left[\begin{array}{cc}5& 1\\ 2& -4\\ & \end{array}\right]+\left[\begin{array}{cc}-5& 2\\ 4& 3\\ & \end{array}\right]·\left[\begin{array}{cc}5& 1\\ 2& -4\\ & \end{array}\right]$

First perform the multiplication between the matrices after it perform addition betwee the resulting matrices.

$\begin{array}{c}AC+BC=\left[\begin{array}{cc}1& 9\\ 26& -8\end{array}\right]+\left[\begin{array}{cc}-21& -13\\ 26& -8\end{array}\right]\\ =\left[\begin{array}{cc}-20& -4\\ 52& -16\end{array}\right]\end{array}$

Step 3. Make the conclusion

As both the sides are not resulting in same matrix which means that the provided equation is false.