How to manage the loss of key resource from the project team in the Matrix Oraganization where project manager reports to functional manager?

This should be handled through the risk planning and should be identified as resource risk. PM needs to list all key resource related roles and risk around it, what if the person playing role is no more, and how to support the business in his/her absence. Good documentation has always been a key remedy for resource related issues. Best way would be meeting key stakeholders and gather details, needed at that key point and plan forward and fill the missing gaps.

Find a replacement withing the organization and work with subject managers within the matrix to assess the feasibilty of allocating his/her services to the project. If that is no possible outsource the role or hire a replacement based on the cost effectiveness and risk assessment of the impact on the project.

The important is not to manage the loss .

As it should be done by replaing him and trying to absorbe the problem that happened .

The important is how to put a mitigation plan for the risk of  loosing a key resource .

This should be done by

1- Good communication plan ensuring that information is not with a single person.

2- Role shadowning , by insuring that each role is covered by atlest1 initial person and1 backup that can replace him in case of emergency ( of course this backup will be acting as  another a primary role in the project ) so each team memeber will have2 roles1 initial &1 backup 

3 -Make sure that project status & risks are shared with functional manager to reduce the chance that they will take such decision that will have bigh impact on your project

In a Weak Matrix Organisation the role of a project manager is that of an expeditor or assistant to the functional manager who has more authority and also controls the resources.

In this case the best option would be for the Project Manager to seek additional resources from the functional manager and in case the functional manager does not cooperate then he should escalate the issue to the Sponsor who is authorised to issue directions to the functional manager to prioritise resources for the project or alternatively to empower the project manager with additional authority over resources.