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Matrix is an open source project that publishes the Matrix open standard for secure, decentralized, real-time communication, and its Apache licensed reference implementations.
It gives you simple HTTP APIs and SDKs to create chatrooms, direct chats and chat bots, complete with end-to-end encryption, file transfer, synchronised conversation history, formatted messages, read receipts and more .
Normally the messages on matrix are end to end encrypted, meaning they can only be read by the recipients with the correct set of keys.
Matrix can integrate with other chat services like IRC, Telegram, Discord etc by the means of bridges. With bridges you can send messages to a telegram group bridged to matrix and they can see and reply back easily.
Matrix has no specific server nor it has a specific app. It is decentralized which makes it safer than the centralized messaging services offered by giant data hogging companies like Facebook, Discord, etc.
Matrix offers VoIP service which means that you can make high quality audio and video calls over the internet and still using an open source and decentralized platform.
Matrix can offer what the user want. If you want unencrypted messages? You can. If you want a different server? You can. If you want a different application? You can. You have the ability to make your own server, even design a whole application if you can, the possibilities are endless.
The concept of Decentralization, Federation and Multiple servers can be a bit difficult to handle by a normal brain, especially if they haven't heard about it before. Also the encryption on matrix is very good which takes us to a famous saying, "Privacy comes with a cost of convenience.". Since every client should be verified with backup keys or use another login to to verify the new one, it becomes a bit difficult than other services like Facebook , Discord which just need you to input your ID and password to login and see all the chats.
Servers are the places where you have your account and where you communicate from. It's like a village where you live, but the servers can go out of fund or the administrator may lose interest and "Your village may get destructed."
Most of the people that use matrix are tech enthusiasts, open source supporters, privacy enthusiasts, and other people (ahmm ! Basically Nerds!). You might not find the people whom you usually talk to in Matrix.
Overall Matrix is very good alternative to other chat services like telegram, signal, messenger, etc. It is very flexible as well as secure which makes it better than all others. And since it's a developing project there are much more features to come. Different clients also are developing "default" features of messaging apps like voice message and other features making easier for the people to shift from their existing platform too if you feel like switching.