Question

Use$A=\left[\begin{array}{cc}2& -1\\ 3& 5\end{array}\right]$,$B=\left[\begin{array}{cc}-4& 1\\ 8& 0\end{array}\right]$ , and$C=\left[\begin{array}{cc}3& 2\\ -1& 2\end{array}\right]$ to determine whether$A\left(BC\right)=\left(AB\right)C$ is true for the given matrices.

Short answer

Yes, the provided equation is true.

Step-by-step solution

Step 1- Determine the value of A(BC)

Substitute the provided matrices into .$A\left(BC\right)$

$A\left(BC\right)=\left[\begin{array}{cc}2& -1\\ 3& 5\end{array}\right](\left[\begin{array}{cc}-4& 1\\ 8& 0\end{array}\right]\cdot \left[\begin{array}{cc}3& 2\\ -1& 2\end{array}\right])$

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

$\begin{array}{c}A\left(BC\right)=\left[\begin{array}{cc}2& -1\\ 3& 5\end{array}\right]\cdot \left[\begin{array}{cc}-13& -6\\ 24& 16\end{array}\right]\\ =\left[\begin{array}{cc}-50& -28\\ 81& 62\end{array}\right]\end{array}$

$\begin{array}{c}A\left(BC\right)=\left[\begin{array}{cc}2& -1\\ 3& 5\end{array}\right]\cdot \left[\begin{array}{cc}-13& -6\\ 24& 16\end{array}\right]\\ =\left[\begin{array}{cc}-50& -28\\ 81& 62\end{array}\right]\end{array}$

– Determine the value of (AB)C

Substitute the provided matrices into .$\left(AB\right)C$

$\left(AB\right)C=(\left[\begin{array}{cc}2& -1\\ 3& 5\end{array}\right]\cdot \left[\begin{array}{cc}-4& 1\\ 8& 0\end{array}\right])\left[\begin{array}{cc}3& 2\\ -1& 2\end{array}\right]$

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

$\begin{array}{c}\left(AB\right)C=\left[\begin{array}{cc}-16& 2\\ 28& 3\end{array}\right]\cdot \left[\begin{array}{cc}3& 2\\ -1& 2\end{array}\right]\\ =\left[\begin{array}{cc}-50& -28\\ 81& 62\end{array}\right]\end{array}$

Step 3- Make the conclusion

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