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.

$c\left(AB\right)=A\left(cB\right)$

Short answer

Yes, the provided equation is true.

Step-by-step solution

Step 1- Determine the value of c(AB)

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

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

First perform the multiplication between the matrices after it multiply the resulting matrix with c.

$\begin{array}{c}c\left(AB\right)=3\left[\begin{array}{cc}-13& -4\\ -8& 17\end{array}\right]\\ =\left[\begin{array}{cc}-39& -12\\ -24& 51\end{array}\right]\end{array}$

– Determine the value of A(cB)

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

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

First perform the multiplication between the matrix B and scalar c after it multiply the resulting matrix with A.

$\begin{array}{c}A\left(cB\right)=\left[\begin{array}{cc}1& -2\\ 4& 3\end{array}\right]\cdot \left[\begin{array}{cc}-15& 6\\ 12& 9\end{array}\right]\\ =\left[\begin{array}{cc}-39& -12\\ -24& 51\end{array}\right]\end{array}$

Step 3- Make the conclusion

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