In the movie, The Matrix is a simulation - a world constructed by an artificial intelligence that is powered by the energy of its human captives. The humans are deceived into believing that they live normal, contributing lives, all the while they're alone in pods having their energy leeched; until one of them - Neo - is freed by a band of underground revolutionaries that believe him to be the "One" (note the anagram) who will defeat the Matrix and free humanity.
In the real world, the earliest reference to the matrix I found was JR Galbraith in 1971. This matrix is an artifact of man, and was intended to help make organizations more efficient. The idea is simple - optimal performance lies in being functionally-integrated vs. vertically-integrated. The specialized teams (Sales, Marketing, Engineering, Design, Research, etc.) will provide their services across product lines, allowing the company to benefit from deep skills and experience by specialty, and the employees to benefit from clearer career paths and professional growth. Makes sense.
Matrix-managed organizations can be more effective and successful if the leaders are able to transcend their team and prioritize the company's success vs. their own. At the top, there are methods like balanced scorecards and dashboards to create an aggregated management view, but it can be difficult for a "line" executive to stand out, and demonstrate their value, ability, and achievement.
How are line executives evaluated? If the organization (like most companies) uses a zero-sum performance model, then the people are measured against each other, despite their work being very different (e.g. Sales is compared to Marketing, Engineering, Design and HR). Sales creates value in a very different way than Marketing, so if Sales exceeds their quota, and Marketing delivers outstanding campaigns, then which manager "wins"? Surely self-interest will prevail, and Sales will want to look better than Mrketing so that it is viewed and rewarded more favorably, which means that in addition to Sales creating value for the company, it has to do it in a way that also shows it to be more valuable than Marketing.
Smart managers will develop their own scorecards to showcase their value (define their own rules of success); if they're "really" smart, some of their metrics will try and take credit for others' success, and thereby increase their odds of a better performance review, and relatively greater recognition and reward.
If a matrixed organization by its nature creates competition between line executives, then why would the C-suite expect there to be cooperation between these competing teams? Look on Monster.com and you'll see a ton of job descriptions that specify the ability to "influence people without authority," or "operate effectively in a matrixed environment," or "work well in cross-team settings." These skills are in demand because managers recognize the value of talent that can get "sister" teams to support and contribute to their activities and success.
A second skill in demand (so much so that it often becomes a discrete role or even a discrete function) is "orchestration." These are the people that develop processes and systems (reports, performance management processes, etc.) that look across organizations to "formalize" horizontal perspective and oversight. In other words, their job is to build patches between the separate horizontal functions.
There is no doubt these skills are necessary and valuable; there is no doubt that specialization is a good thing - the better you are at something, the better you are at it. There is also no doubt that a well-run matrixed organization can be more efficient than the alternative. Why then is there so much angst/difficulty/stress about getting things done, being more agile, and being more responsive to the changing market in matrixed organizations? How do we make the matrix work?
I think it's simple, and the movie can teach us something. To prevail against the AI and win humanity's freedom, Neo and the resistance had to set aside local priorities and ego and work together towards the common purpose. Their measure of success was freedom for all people, and the price of failure was death (or re-absorption into the matrix). First, it was easy to keep score; and second and most critically, the penalty for failure and the rewards of success were shared equally by all participants. That's what kept them motivated, and what helped them persevere when things looked most dire.
Organizations (matrixed or otherwise) are more successful and stronger when everyone's success is based on the company's success; and the knowledge that if the company fails, everyone takes it on the chin. Everyone's success should be a function of the company's success, and everyone should similarly know what it's like to lose.
For example, we will all get a share (equal amount or percentage of salary) of "profits" when we succeed (no other bonus), and get no bonus when we fail. The rule should be applied universally and irrespective of whether you did all the work, but your colleague in the next cubicle did nothing but watch YouTube videos all day. This will put all sorts of positive pressure on the company and its people - to want everyone to be productive (and "help" or "encourage" those that aren't); to hire well (so critical); and of course to work across teams (I need to help Sales, and Marketing and Design so that the company is successful and that we in Engineering helped make us so).
In the end, it's not whether you work in a matrix (or The Matrix), but rather how you get compensated. A matrixed approach will not guarantee success, but the right measurement scheme will.
Shared vestedness begets participation, which begets teamwork, which begets shared success.