If using a matrix, you would need one with EDID management control of the setup and all displays would need to have matching resolution. The headache is the audio and the lowest common denominator factor if you don't have proper EDID control. This is a feature on better HDMI matricies, but not on cheaper ones. Also, it isn't always straightforward how to setup EDID on a matrix.
So, this all said, you can do it if that's the end goal.
Sources would be plugged into the HDMI matrix and outputs to at least two TVs would be connected, then the third TV can be connected to the AV receiver, or the third TV can be connected directly to the matrix, and the AV receiver can be connected to the matrix separately for audio playback only.
I use a 16x16 HDMI matrix in my home. It includes breakaway audio and I use DSP input cards which downmix surround sound to stereo for rooms that need it, though those rooms must use separate speakers from the TVs. All of the inputs are setup to support full surround sound, and I pull a stereo downmix off for my non-surround sound rooms. The two rooms that have AVRs in them, get a output feed off the matrix that includes both audio and video and the video is passed through to the TV while the AVR handles audio to the room.
If you need 4K video, this will also increase the cost a fair bit, but it is out there. Better control system companies like Crestron and Control4 do this on a regular basis. I haven't looked into other companies with this capability. Heck, I only have 'heard' that Control4 does it. I KNOW Crestron does it as that's what I use and part of my career.
Taking a brief look online, something like this may do the trick:
Offering transmission of 18Gbps HDMI content up to 35m/115ft by pairing HDBaseT with WyreStorm’s powerful lossless compression algorithm, the MXV-0606-H2A v3 brings the familiar power of the WyreStorm H2-series to a compact 1U chassis. Unlike competing compression methods such as CSC, which...
www.wyrestorm.com