Why does the order of matrix multiplication matter?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, the order of matrix multiplication matters because matrix multiplication is not commutative. Unlike the multiplication of ordinary numbers, where $a b = b a$ always holds true, the product of two matrices $A$ and $B$ generally results in different outcomes depending on whether you calculate $A B$ or $B A$ .1

The definition of matrix multiplication requires that the number of columns in the first matrix must equal the number of rows in the second. Consequently, if the dimensions of $A$ and $B$ are not compatible in both directions, one product may be mathematically defined while the other is not. Even when both $A B$ and $B A$ are defined—such as with square matrices of the same order—the resulting elements are determined by the sum of products of specific rows and columns, which change entirely when the order of the matrices is swapped.1

While matrix multiplication is associative and distributive, the failure of the commutative law is a fundamental characteristic of matrix algebra.1