In mitochondria, ATP synthesis occurs through a process called oxidative phosphorylation, which takes place in the inner membrane. The inner membrane of the mitochondria contains a series of protein complexes called the electron transport chain. This chain consists of several proteins that transfer electrons from one molecule to another, ultimately creating a proton gradient across the inner membrane.

The production of ATP occurs in this inner membrane because the proton gradient drives the synthesis of ATP by the enzyme ATP synthase. ATP synthase acts as a molecular motor and harnesses the energy from the proton gradient to add a phosphate group to ADP, converting it into ATP.

Although the outer membrane and matrix of the mitochondria play important roles in other cellular processes, such as regulating the flow of molecules in and out of the mitochondria and housing the mitochondrial DNA, they are not directly involved in the ATP synthesizing chemical reaction.

Therefore, the correct answer is option 3, the inner membrane.