Why is the matrix multiplication defined as it is? [duplicate]

Matrix multiplication is defined as:

Let $A$ be a $n \times m$ matrix and $B$ a $m\times p$ matrix, the product $AB$ is defined as a matrix of size $n\times p$ such that $(AB)_i,_j = \sum\limits_{k=1}^mA_i,_kB_k,_j$.

For what good reason did mathematicians define it like this?

Solution 1:

A matrix is nothing but a particular representation of a linear map (with respect to a choice of basis in source and target space). The formula is what results (naturally) if you look at the composition of such maps and write them down using a matrix.

Solution 2:

One reason is that it gives you associativity with a vector: If $A$ and $B$ are matrices and $x$ is a vector, then

$$ (AB)x = A(Bx) $$

ETA: This doesn't say anything more than Thomas's answer, by the way; I thought it might help if it were presented in a more tangible way, though.