Remove index.php from Laravel 4 URL and Rewrite www to non-www - apache

Question

I have tried several ways to remove index.php from my URL (http://www.example.com/index.php/login), as well as redirecting www to the non-www URL.
Here is my default Apache VirtualHost file:
<VirtualHost *:80>
ServerAdmin webmaster#example.com
ServerName example.com
ServerAlias www.example.com
DocumentRoot /var/www/public/
<Directory />
Options FollowSymLinks
AllowOverride All
</Directory>
<Directory /var/www/public/>
Options Indexes FollowSymLinks MultiViews
AllowOverride All
Order allow,deny
allow from all
</Directory>
ScriptAlias /cgi-bin/ /usr/lib/cgi-bin/
<Directory "/usr/lib/cgi-bin">
AllowOverride None
Options +ExecCGI -MultiViews +SymLinksIfOwnerMatch
Order allow,deny
Allow from all
</Directory>
ErrorLog ${APACHE_LOG_DIR}/error.log
# Possible values include: debug, info, notice, warn, error, crit,
# alert, emerg.
LogLevel warn
CustomLog ${APACHE_LOG_DIR}/access.log combined
</VirtualHost>
Now I have added the following in to this file:
Redirect 301 / http://example.com
But this has not worked, I just get a redirect loop message. I also have the following .htaccess:
<IfModule mod_rewrite.c>
Options -MultiViews
RewriteEngine On
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^ index.php [L]
</IfModule>
To which I've added:
RewriteBase /
RewriteCond %{HTTP_HOST} ^www\.(.*)$ [NC]
RewriteRule ^(.*)$ http://%1/$1 [R=301,L]
But this does not work either.
The other issue is that for all of my routes I have to include index.php, like http://example.com/index.php/test
It's as if my .htaccess file is being ignored.
The server is a DigitalOcean Droplet running Ubuntu 13.04 x64. It has PHP 5.4.9 and Apache 2.2.22.
Any help would be very much appreciated. Thanks.

Answer 1

Thanks to anlutro in #laravel IRC chat, he pointed out that mod_rewrite may not be enabled. Usually this is on by default in all the servers I've ever worked with so it didn't click, as I've never had to enable it but on this DigitalOcean server, it wasn't.
I used:
a2enmod rewrite to enable it and then I restarted apache. All is good.
Cheers