GNU Privacy Guard vs Matrix

GPG works on OS X, Linux, and Windows with extensive selection of wrappers.

GPG supports public key cryptography (RSA EIGamal, DSA), symmetrical key algorithms (Blowfish, AES, IDEA, etc), cryptographic hash functions (RIPEMD, SHA), and compression (ZIP, ZLIB, BZIP2).

Open source (the free version of openPgp), other similar products are offered for other platforms (Gpg4win for Windows for example). It is the most secure way to send emails today, close to military encryption. Some negative points: not all mail clients support it and when a key is not only on your hard disk but also on a server (pgp.mit.edu for example), it is complicated to remove it.
Last positive point: it can also be used other than for emails, to encrypt and sign documents. Bitcoin releases are signed with the GPG keys of the developers ;)

Allows for encrypted communication.

With GPG you can encrypt you whole volume or files individually.

GPG keys by default expire after a set amount of time. The amount can be changed and this feature can be turned off.

GPG can be installed as a command line tool, or you can choose between several different GUI frontends available for it.

GPG is the oldest and most reliable encryption software available.

Matrix is an open standard, defining simple HTTP APIs so that devs can easily write their own clients, bots, bridges or servers. You're not locked into a specific set of implementations.

Matrix has bridges to IRC (freenode, moznet, oftc, snoonet etc), Slack, Gitter, Rocket.Chat, XMPP, SMS, SIP and others. The point is to 'matrix' all the different networks out there into one single decentralised network.

Riot.im is the easiest way to use Matrix, with great clients for Web, iOS and Android (and Fdroid).

Matrix is decentralized, there's no one central point that the information goes through and so no once central point of failure or control.

Matrix aims to "provide an analogous ecosystem to email - one where you can communicate with pretty much anyone, without caring what app or server they are using" using a neutral identity system.

Matrix is designed to support Instant Messaging, VoIP/WebRTC signalling (voice and video) and Internet of Things communication.

Riot.im includes an app store with integrations for Github, JIRA, Jenkins, Giphy etc - and anyone can add more via Matrix.

Which can be found here.

Matrix is designed to support Instant Messaging, VoIP/WebRTC signalling and Internet of Things communication and allows cross-communication between those services. Meaning one person could be using IRC and another Slack for IMs, or one person could be using Skype and another Google Hangouts for videoconferencing.

Matrix features end-to-end encrypted chats which are syncronized accross all your devices.

E2E is implemented in the matrix-js-sdk library and Riot.im client.