I suggest you use 'perl -d (yourscript.pl)' to take a look (>x $matrix_final) at what those two references are made of.
Once you do so, it will become apparent to you how to do all three of the things you want.
Both $matrix_final and $matrix transpose are simply references to arrays of arrays. The basic perldoc (perldsc I think) will show you how to do what you want to do... .

That should get you started. With a little work you will have what you want shortly...

Hope that is helpful...

Update:

Added link to perldsc to be a bit more helpful...

Update: rev b Looking at Math::MatrixReal it seems that most of what you want to do is actually supplied by the module...
It will be helpful to actually read that...
In particular, $new_matrix = Math::MatrixReal->new_from_string($string); This method allows you to read in a matrix from a string (for instance, from the keyboard, from a file or from your code).