Mozilla Thunderbird
Thunderbird is a free, open-source email, calendar, and chat application which runs on multiple platforms. It supports multiple instant messaging protocols, including IRC, XMPP, and Matrix.
Thunderbird is available for download and the source code is available. Additionally there is documentation for getting started with contributing. Please file any bugs found.
Matrix support in Thunderbird is enabled by default in Daily builds (as of Thunderbird 91), but can be enabled in beta and release builds by setting chat.prpls.prpl-matrix.disable to false in the advanced config editor (>Tools >Options >Advanced >Config Editor).