alternately, to avoid len(file1) x len(file2) loops,

use autodie;
use warnings;
use strict;
my (@data1, @data2) = ();
my ($fh, $l, $n);
open $fh, "<", $ARGV[0];
while(<$fh>) {
    ($l, $n) = split;
    push @data1, [ $l, $n ];
}
close($fh);
open $fh, "<", $ARGV[1];
while(<$fh>) {
    ($l, $n) = split;
    push @data2, [ $l, $n ];
}
close($fh);

foreach my $row1 ( @data1 ) {
    foreach my $row2 ( @data2 ) {
        my ($l1, $n1, $l2, $n2) = (@$row1, @$row2);
        my $match = $l1 eq $l2;
        print "$l1 from FILE1 with number $n1 and $l2 from FILE2 with 
+number $n2" . ($match ? '' : " DO NOT") . " match\n";
    }
}
__END__
A from FILE1 with number 1_1 and A from FILE2 with number 2_1 match
A from FILE1 with number 1_1 and B from FILE2 with number 2_2 DO NOT m
+atch
A from FILE1 with number 1_2 and A from FILE2 with number 2_1 match
A from FILE1 with number 1_2 and B from FILE2 with number 2_2 DO NOT m
+atch
B from FILE1 with number 1_3 and A from FILE2 with number 2_1 DO NOT m
+atch
B from FILE1 with number 1_3 and B from FILE2 with number 2_2 match
C from FILE1 with number 1_4 and A from FILE2 with number 2_1 DO NOT m
+atch
C from FILE1 with number 1_4 and B from FILE2 with number 2_2 DO NOT m
+atch
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this will save a lot of time if the files are significantly larger than 4 and 2 lines, respectively. though it will end up using more memory...