Problem

You want to multiply a pair of two-dimensional arrays. Mathematicians and engineers often need this.

Solution

Use the PDL modules, available from CPAN. PDL is the Perl Data Language  - modules that give fast access to compact matrix and mathematical functions:

use PDL;
# $a and $b are both pdl objects
$c = $a * $b;

Alternatively, apply the matrix multiplication algorithm to your two-dimensional array:

sub mmult {
    my ($m1,$m2) = @_;
    my ($m1rows,$m1cols) = matdim($m1);
    my ($m2rows,$m2cols) = matdim($m2);

    unless ($m1cols == $m2rows) {  # raise exception
        die "IndexError: matrices don't match: $m1cols != $m2rows";
    }

    my $result = [];
    my ($i, $j, $k);

    for $i (range($m1rows)) {
        for $j (range($m2cols)) {
            for $k (range($m1cols)) {
                $result->[$i][$j] += $m1->[$i][$k] * $m2->[$k][$j];
            }
        }
    }
    return $result;
}

sub range { 0 .. ($_[0] - 1) }

sub veclen {
    my $ary_ref = $_[0];
    my $type = ref $ary_ref;
    if ($type ne "ARRAY") { die "$type is bad array ref for $ary_ref" }
    return scalar(@$ary_ref);
}

sub matdim {
    my $matrix = $_[0];
    my $rows = veclen($matrix);
    my $cols = veclen($matrix->[0]);
    return ($rows, $cols);
}

Discussion

If you have the PDL library installed, you can make use of its lightning-fast manipulation of numbers. This requires far less memory and CPU than Perl's array manipulation. When using PDL objects, many numeric operators (such as + and *) are overloaded and work on an element-by-element basis (e.g., * is the so-called scalar multiplication operator). To get true matrix multiplication, use the overloaded x operator.

use PDL;

$a = pdl [
    [ 3, 2, 3 ],
    [ 5, 9, 8 ],
];

$b = pdl [
    [ 4, 7 ],
    [ 9, 3 ],
    [ 8, 1 ],
];

$c = $a x $b;  # x overload

If you don't have the PDL library, or don't feel like pulling it in for a small problem, you can always do the work yourself the good old-fashioned way.

# mmult() and other subroutines as above

$x = [
       [ 3, 2, 3 ],
       [ 5, 9, 8 ],
];

$y = [
       [ 4, 7 ],
       [ 9, 3 ],
       [ 8, 1 ],
];

$z = mmult($x, $y);

See Also

The documentation with the CPAN module PDL