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.

$AC+BC=(A+B)C$

Short answer

The provided equation is true.

Step-by-step solution

Step 1- Determine the value of AC+BC

Substitute the provided matrices into $AC+BC$

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

First perform the multiplication after it perform addition.

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

– Determine the value of (A+B)C

Substitute the provided matrices into $(A+B)C$

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

First perform the addition after it multiplication.

$\begin{array}{c}(A+B)C=\left[\begin{array}{cc}-4& 0\\ 8& 6\end{array}\right]\cdot \left[\begin{array}{cc}5& 1\\ 2& -4\end{array}\right]\\ =\left[\begin{array}{cc}-20& -4\\ 52& -16\end{array}\right]\end{array}$

Step 3- Make the conclusion

As both the sides results in the same matrix which means that the provided equation is true.