Question

In general, matrix multiplication is not commutative. Show by counterexample that this is true. Then give an example of when matrix multiplication is commutative. Explain.

I have no idea where to even begin....

Answer 1

\[\left[\begin{matrix}1&2\\3&4\end{matrix}\right]\cdot\left[\begin{matrix}1&3\end{matrix}\right]\not=\left[\begin{matrix}1&3\end{matrix}\right]\cdot\left[\begin{matrix}1&2\\3&4\end{matrix}\right]\]
In this case, the LHS is undefined because the number of columns in the first matrix is not equal to the number of rows in the second.

And another example, even when the number of columns matches the number of rows:
\[\left[\begin{matrix}1&2\\3&4\end{matrix}\right]\cdot\left[\begin{matrix}1&3\\2&0\end{matrix}\right]\not=\left[\begin{matrix}1&3\\2&0\end{matrix}\right]\cdot\left[\begin{matrix}1&2\\3&4\end{matrix}\right],\text{ since}\\
\left[\begin{matrix}5&3\\11&9\end{matrix}\right]\not=\left[\begin{matrix}10&14\\2&4\end{matrix}\right]\]

Answer 2

Thank you!