Q. 4 - Precalculus Enhanced with Graphing Utilities

Question

True or False 

Matrix multiplication is commutative.

Step-by-Step Solution

Answer

False.

Step 1. Given Information

Two matrices A & B.

Step 2. Calculation

$$\begin{gathered} Consider two matrix A \& B \\ let A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right] \\ B=\left[\begin{array}{cc}-1& 2\\ 1& -3\end{array}\right] \\ AB=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}-1& 2\\ 1& -3\end{array}\right] \\ =\left[\begin{array}{cc}-1+2& 2-3\\ -3+4& 6-12\end{array}\right] \\ =\left[\begin{array}{cc}1& -1\\ -1& -6\end{array}\right] \\ \hspace{0.17em}BA=\left[\begin{array}{cc}-1& 2\\ 1& -3\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right] \\ =\left[\begin{array}{cc}-1+6& -2+8\\ 1-9& 2-12\end{array}\right] \\ =\left[\begin{array}{cc}5& 6\\ -8& -10\end{array}\right] \\ AB\ne BA \end{gathered}$$