In the following, multiply the first two matrices together. Click on the = to check your result. Then multiply the result with the third matrix.
Now do the problem again, but this time start with the last two matrices.
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.
Say that A 5×5 B ?×? C 3×4 = D ?×?
What are the dimensions of B and D ?