Show that Matrix Multiplication is Associative

[Saladsamurai]

Homework Statement

Show that (AB)C=A(BC)

I am just trying to do this to try to gain some experience with problems like this. I saw in my text that they did a similar example for distributivity using the definition of matrix multiplication, so I thought I could use that approach.

The Attempt at a Solution

Let the (i,j)-entry of A be given by a_ij
Let the (i,j)-entry of B be given by b_ij
Let the (i,j)-entry of C be given by c_ij

Then the (i,j)-entry of (AB) is given by

$$\sum_{k=1}^na_{ik}b_{kj}$$

Here is where I get lost. I was thinking of then writing that the (i,j)-entry of (AB)C would be given by

$$\sum_{k=1}^n(a_{ik}b_{kj})c_{kj}$$but I don't think that this works...and I am not sure why or why not

Any hints?

Thanks!

 

[Office_Shredder]

You have the right idea, but your element of (AB)C is incorrect. Each entry in AB is going to be a summation, and then each entry in (AB)C should be a summation in which each term contains a summation (specifically, the piece from AB is going to be a summation). So your final equation for the (i,j)th entry of (AB)C should contain a double summation.

Just walk through the definition of matrix multiplication real carefully and you should be able to get it