Question

Explain why, in most cases$(A+B)C\ne CA+CB$

Short answer

The distributive property is not always true for matrices.

Step-by-step solution

Step 1- Observe the provided information

The provided information indicates towards distributive property.

– Use the observation

Solving matrix by distributive property$a(b+c)=ab+ac$will lead to different matrix on both the sides most of the times.

For example,

Considerlocalid="1648734720106" $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$localid="1648734733768" $B=\left[\begin{array}{cc}2& 2\\ 1& 2\end{array}\right]$and localid="1648734748052" $C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Determine the value of $(A+B)C$

localid="1648734790943" $\begin{array}{c}(A+B)C=\left[\begin{array}{cc}3& 4\\ 3& 3\end{array}\right]\cdot \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\\ =\left[\begin{array}{cc}4& 3\\ 3& 3\end{array}\right]\end{array}$

Determine the value of $CA+CB$

localid="1648735012033" $\begin{array}{c}CA+CB=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]+\left[\begin{array}{cc}1& 2\\ 2& 2\end{array}\right]\\ =\left[\begin{array}{cc}3& 3\\ 3& 4\end{array}\right]\end{array}$

Both the matrices are not the same.

Step 3- Write the conclusion

The using of observation results in that the distributive property is not always true for the matrices.