A quicker alternative, which I found out by chance a few months ago.

Simpler without ::Simple. :-)

#!/usr/bin/perl -w
use strict;
use Spreadsheet::ParseExcel::Simple;
use Spreadsheet::WriteExcel;           # <--- 

my $xls = Spreadsheet::ParseExcel::Simple->read('old.xls');
my @data;
for ($xls->sheets) {
   while ($_->has_data) {
       push @data, [$_->next_row];
   }
}

my $ss1 = Spreadsheet::WriteExcel->new('new.xls');
my $ws = $ss1->add_worksheet('transposed');
$ws->write('A1', \@data);
# Yes. IT IS transposed!
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As a side note, in your example, instead of Math::Matrix you can use tye's Algorithm::Loops.

use Algorithm::Loops qw(MapCar);
# ....

$ss->write_row($_) for MapCar {[@_]} @data ;
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