Non-Commutativity of Matrix Multiplication

Linear Algebra (Matrices) · Axiom Academy

LESSON Non-Commutativity of Matrix Multiplication Unlike ordinary numbers, the order of a matrix product matters: in general . 1. Same Matrices, Opposite Order, Different Result Take the two matrices below. We multiply them both ways at once — top row computes AB , bottom row computes BA — filling each result cell as a row of one matrix meets a column of the other. Watch the two results land on different grids . 2. Why: A Product Is "Do One Transform, Then the Other" A matrix is a transformation of the plane, and a product applies them right-to-left : means "rotate , then shear." Below, the same square is fed through a rotation and a shear in both orders. The final shapes don't match — and that mismatch is . 3. Sometimes One Order Isn't Even Defined Order can matter even more drastically. For a product to exist, the inner dimensions must match : the columns of the left factor must equal the rows of the right. Watch the dimension blocks try to lock together in each order — one fits, one leaves a gap. Here A is and B is . Then AB is defined and produces a matrix — but BA is impossible, because B has 4 columns while A has only 2 rows. One order works; the other doesn't exist. 4. The Special Cases That DO Commute Non-commutativity is the default , not a law without exceptions. Some pairs always commute — and each is worth recognizing on sight. The animation checks four of them, collapsing each AB and BA onto the same result. Any matrix commutes with I : .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.