Associative

In the following, multiply the first two matrices together. Click on the = to check your result. Then multiply the result with the third matrix.

Problem		Partial Result		Result
1 -1 2 3 -2 1 0 2 -1 0 1 1		? ? ? ? -1 0 1 1		? ? ? ?

Now do the problem again, but this time start with the last two matrices.

Problem		Partial Result		Result
1 -1 2 3 -2 1 0 2 -1 0 1 1		1 -1 2 3 ? ? ? ?		? ? ? ?

The final answer is the same for both ways of doing the problem. This demonstates that matrix multiplication is associative:

(AB)C  =  A(BC)

Of course, the inner dimensions of A and B must be the same, and the inner dimensions of B and C must be the same. Usually a product of three matrices is written ABC.