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

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

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.