Question: Please keep in mind that I know nothing about Matlab.

Matlab Builder JA lets developer build Matlab applications and export them into Java jars. That's great, I just have to produce a jar and I can then use it from other java code.

Does anyone know how much the single jar packaging module cost?

Is there any free version or similar freeware product?

Is there any other way to achieve the same thing -Using Java to pass inputs to Matlab and getting an output back without worrying about anything else- with standard Matlab/Java?

Answer: 
MATLAB Builder JA for Java is currently £3,150 for an individual commercial license, and requires MATLAB Compiler, which is currently £3,850 for an individual commercial license. I'm in the UK so can't get pricing in other currencies, but you can get your local prices from the following links.

[Pricing for MATLAB Builder JA](http://www.mathworks.com/products/javabuilder/pricing_licensing.html)

[Pricing for MATLAB Compiler](http://www.mathworks.com/products/compiler/pricing_licensing.html)

Contrary to Birdasaur's answer, the products (and the deployed components) are not supported on Solaris - MATLAB itself has not been supported on Solaris [since R2010a](http://www.mathworks.com/support/sysreq/roadmap.html). You can also deploy the generated .jar files to as many sites as you like. Individual licenses can be either assigned to a named individual, in which case only that developer can use the product; or to a specific machine, in which case any developer can use it as long as they are at the console of the machine (not remotely logged in).

MATLAB also has an undocumented interface called JMI (Java MATLAB interface) that you can use to call MATLAB directly from Java. Take a look at [matlabcontrol](http://code.google.com/p/matlabcontrol/). However, this requires a live copy of MATLAB for the deployed application.
You should probably contact MathWorks about licensing. As this is fairly high end functionality I would speculate that it is quite expensive.

You should probably take a look at [Octave](http://www.octave.org) which is licensed under GNU GPL. Also, there are also a wide variety of wrappers around Matlab, such as [MLabWrap](http://mlabwrap.sourceforge.net/), however they require a Matlab version installed, so it would not work for redistribution or anything.