In matrix organization project managers and functional heads may or may not have a lot of power. It depends on the "Strong Matrix" or "Weak Matrix" on whether an outcome would be successful. Give two scenarios in which a project was successful in an matrix organization or unsuccessful.
The first successful example of a "matrix" organization I have seen was when I was a co-op at an ultrasound transducer manufacturing facility. The project was a migration of upstream components to a supplier. In this example, manufacturing (which acted in more of a functional manner) and the projects organization (which acted in more of a project manner) were able to communicate well and not "silo" while also not fighting to divert work to each other.
The second successful example of a "matrix" organization was the execution of a plant startup. Here, manufacturing, supply chain, the business operating unit, and the project managers all had equal say on the execution of the project. The infighting among the groups I witnessed previously was absent, leading to the project progressing much faster than what I have seen previously.
In my personal experience I have only experienced a "Weak Matrix" organization, so I can only speak to that scenario from experience, but I can assume how it might work in a "Strong Matrix" organization.
From my own experience, I was part of a project team as a quality representative. This project went very well because of how experienced both the project manager and my department's functional lead are. Although it was clear that the functional lead had more power in deciding exactly how much of my time and effort I would be resourcing to this project, they were able to communicate their perspectives very well to each other. The project manager was effective in conveying how important this project is, and the functional lead was very aware of the other work that I was doing. The project manager was also effective in communicating the project clearly to the project team and explicitly outlining the timeline and deliverables from each member. This made it very simple to complete the project in a timely manner.
In a strong matrix, I would assume that a lot of the staff members under each functional lead might be very overburdened. If project managers are able to endlessly demand the use of people from each department, I would imagine that they would not really care about how many other projects you are handling, making way to poor time management and prioritization by individual members of each project team.
I will say that I must be a little biased because this is based off of my own experiences and I have no idea how the concept of a Strong Matrix would translate into real life, this is only based on my assumptions.