What is the Matrix ?

What is the Matrix? 
In the end what you are first and formostly experiencing & battling with is your own belief system. Separating mind from matter, 'the world around you', and the physical was part of the creation of the illusion which would become the prison.

Therefore what the Matrix really boils down to is a belief that there's a Matrix/prison outside you, separate from you, toward which you are powerless.
At 2:48 Morpheus clearly says "It is a prison for your mind".
And in the beginning (@1:16) Morpheus gives it away when he asks Neo if he believes in fate, or thus not being in control of his own destiny.

The matrix is your belief there is a matrix outside of you which is in control of you. 

A thought process which keeps you focused on something outside of you and keeps you fighting 'it' or attaching to 'it', not seeing your own thought process/belief itself.

People locked up in physical prisons just make you believe the prison is 'out there', just like addicts,  due to which you will never figure out that your own mind is being used by you to imprison you ...that you are a slave of yourself. That the warden of your prison is you yourself.

Nobody will fight u harder than yourself for freeing yourself.
That's now the best place to hide the prison, from ourselves, i.e., in your own mind, and make ourselves the very warden of our own prison.
No other prison could in the end be able to sustainably keep everyone locked up.