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.

$ABC=CBA$

Short answer

The provided equation is false.

Step-by-step solution

Step 1- Determine the value of ABC

Substitute the provided matrices into ABC.

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

First perform the multiplication between the matrices A and B after it multiply the resulting matrix with C.

$\begin{array}{c}ABC=\left[\begin{array}{cc}-13& -4\\ -8& 17\end{array}\right]\cdot \left[\begin{array}{cc}5& 1\\ 2& -4\end{array}\right]\\ =\left[\begin{array}{cc}-73& 3\\ -6& -76\end{array}\right]\end{array}$

– Determine the value of CBA

Substitute the provided matrices into CBA.

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

First perform the multiplication between the matrices C and B after it multiply the resulting matrix with A.

$\begin{array}{c}CBA=\left[\begin{array}{cc}-21& 13\\ -26& -8\end{array}\right]\cdot \left[\begin{array}{cc}1& -2\\ 4& 3\end{array}\right]\\ =\left[\begin{array}{cc}31& 81\\ -58& 28\end{array}\right]\end{array}$

Step 3- Make the conclusion

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