I am studying a math lesson where we want to demonstrate that, given three matrices A, B and C, (AB)C = A(BC).
Here is a part of the lesson that expresses the (AB)C matrix in function of i and j :
While I do understand the first line, I don't get at all the meaning of the second one !
Why are we doing the sum of sums here ?
For the first line, that was easy, k corresponds to the row of (AB) and to the column of C.
In the second line, I don't understand what k and l correspond to... more precisely, I don't understand why we have two sums, and why l and k are used at those places.
To be clear : I don't understand anything in the second line.
Could you help me please ? Thank you.
The new sum is definition of the $i,k$ entry of $AB$.