# Why is matrix multiplication not commutative?

First off, if we aren't using square matrices, then we couldn't even try to commute multiplied matrices as the sizes wouldn't match. But even with square matrices we don't have commutitivity in general. Let's look at what happens with the simple case of $2 \times 2$ matrices.
Given $A = \left(\begin{matrix}{a}_{11} & {a}_{12} \\ {a}_{21} & {a}_{22}\end{matrix}\right)$ and $B = \left(\begin{matrix}{b}_{11} & {b}_{12} \\ {b}_{21} & {b}_{22}\end{matrix}\right)$
$A B = \left(\begin{matrix}{a}_{11} {b}_{11} + {a}_{12} {b}_{21} & {a}_{11} {b}_{12} + {a}_{12} {b}_{22} \\ {a}_{21} {b}_{11} + {a}_{22} {b}_{21} & {a}_{21} {b}_{12} + {a}_{22} {b}_{22}\end{matrix}\right)$
$B A = \left(\begin{matrix}{a}_{11} {b}_{11} + {a}_{21} {b}_{12} & {a}_{12} {b}_{11} + {a}_{22} {b}_{12} \\ {a}_{11} {b}_{21} + {a}_{21} {b}_{22} & {a}_{12} {b}_{21} + {a}_{22} {b}_{22}\end{matrix}\right)$
Notice that these are not going to be the same unless we make some very specific restrictions on the values for $A$ and $B$. Because you're taking the rows from the first matrix and multiplying by columns from the second, switching the order changes the values that are going to occur for any given element.