10   0  20
20  40  10
 0  20  10

-2   2 
 1   1
 2  -2

1   0   2
2   4   1
0   2   1

-2   2 
 1   1
 2  -2

2  -2
2   6
4   0

1  -1
1   3
2   0

Associative

Multiply the first two following 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 by multiplying 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 the fact that matrix multiplication is associative:

(AB)C = A(BC)

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