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Which arithmetic laws hold for matrix multiplication?

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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, matrix multiplication follows several fundamental arithmetic laws, though it differs from ordinary multiplication in significant ways.

When the dimensions of the matrices allow for the operations to be performed, the following laws hold true:

Associative Law: The grouping of matrices does not affect the result of the product, expressed as $A (B C) = (A B) C$ .1
Distributive Law: Matrix multiplication is distributive over addition, meaning $A (B + C) = A B + A C$ and $(B + C) A = B A + C A$ .1

It is important to note that the commutative law—which states that $a b = b a$ in ordinary arithmetic—does not generally hold for matrix multiplication; in most cases, $A B \neq B A$ . While there are specific exceptions, such as when multiplying certain diagonal matrices, commutativity is not a general property of matrix systems.21