My evaluation and opinion:
MATRIX is a promising new protocol for messengers that still has some teething troubles. The worst thing is that there is no #PrivacyByDefault and you have to enable end-to-end encryption first. There are client implementations without encryption that should not be used. The most common client is RIOT and it supports all important features. RIOT can be installed via f-droid.org and supports encrypted group chats as well as video chats and file exchange.
The infrastructure of Matrix is decentralized. A major advantage over XMPP are federated group chats. This means that if a country wants to censor a group chat via XMPP, it can take the corresponding server off the network or block it. This is not possible with Matrix, because the group chat is still available on other Matrix servers. This makes the infrastructure very resistant to censorship of single countries.
The MATRIX server supports bridges to IRC, XMPP, Telegram and Signal (and other). This means that you can join group chats with these other messengers. One of the current development problems is the performance and scaling with many users. It is excellent that France is using MATRIX for its communication within the administration. Germany uses MATRIX for its military.
MATRIX can already be used productively if you make sure that the encryption is enabled.
Complete open source project (free software) and available on f-droid.org
Decentralized and censorship resistant. Video chats, file exchange and bridges to other messengers. The software is available for all major platforms: iOS, Android, Linux and Windows.
There is an end-to-end encryption if you enable it, which is not the default setting.
Performance and scaling problems with many users. Complicated server installation. Not #PrivacyByDefault. Interaction problems with different servers when shared group chats are used. They are often not listed in the search.