Q28. - Algebra 2

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.

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

Step-by-Step Solution

Answer

Yes, the provided equation is true.

Step 1- Determine the value of c ( A B )

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. 

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

Step 2 – Determine the value of A ( c B )

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.