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Why is matrix multiplication not always commutative?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In algebra, matrix multiplication is not commutative because the operation depends on the specific order of rows and columns, unlike the multiplication of ordinary numbers where the commutative law ( $a b = b a$ ) always holds.12

The product of two matrices $A$ and $B$ is defined by taking the dot product of the rows of the first matrix with the columns of the second. Because this process relies on the specific dimensions and arrangement of elements within each matrix, reversing the order ( $B A$ ) typically results in a different outcome or may even be undefined if the dimensions do not align correctly. While matrix multiplication is associative and distributive, it does not satisfy the commutative property in general systems.12

An exception exists for diagonal matrices, where multiplication is commutative, but this is a specific case rather than a general rule for all matrices.1