Why is matrix multiplication not commutative?

Question

Abigail Allen · Accepted Answer

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 #2xx2# matrices. Given #A = ((a_11, a_12),(a_21,a_22))# and #B = ((b_11, b_12),(b_21,b_22))# #AB = ((a_11b_11 + a_12b_21, a_11b_12 + a_12b_22),(a_21b_11+a_22b_21, a_21b_12+a_22b_22))# #BA = ((a_11b_11 + a_21b_12, a_12b_11 + a_22b_12),(a_11b_21+a_21b_22, a_12b_21+a_22b_22))# 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.

Eva Adamson · Answer

undefined Matrix multiplication is not commutative because the order in which matrices are multiplied affects the result. Specifically, if $A$ and $B$ are matrices, then in general $AB \neq BA$. This non-commutativity arises because the number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be defined. When multiplying matrices in a different order, the dimensions no longer match, leading to a different result.