For many years, I too have suggested people look at PDL, but it's only now reflecting on zentara's suggestion that I see PDL could really use some example code up on perlmonks. Unfortunately, I'm not a big PDL user, so you might want to jump onto the PDL mailing list or peruse the Quantified Onion for more knowledgeable answers.

Taking a stab at an example, would the PDL code look something like this?

use PDL;

# amino acids in this order in the reaction rates pdl
# A R N D C Q E G H I L K M F P S T W Y V B Z X
$rates = pdl [
    [ 5, -2, -1 ],
    [ -2, 7, -1 ],
    [ -1, -1, 7 ],
];

$multiplied_rates = $rates x $rates;
print $multiplied_rates;
[download]

My calculation by hand seems to agree with the first two values. The rest is left as an exercise for the user.

Looks a lot easier than the OP's code, if you can overcome the fear of the new. And it's kinda cool :)

Edit:If I'd checked before posting, I would have linked to The Perl Data Language (PDL): A Quick Reference Guide or RFC: Getting Started with PDL (the Perl Data Language) both of which have lots of examples.