Sounds are formally represented as matrices of binary or multi-valued features (e.g., [+voice], [+anterior]). This allows systematic statement of which feature combinations form natural classes and how features change in phonological processes.
From your work with phonological features, you know that sounds are not atomic — they are bundles of articulatory and acoustic properties. A feature matrix is simply the formal notation that makes this explicit: instead of writing a sound as a symbol (like /p/ or /d/), you write it as a column of feature-value pairs, each feature set to + or − depending on whether the sound has that property. The matrix for /b/, for example, would show [+voice], [+labial], [−nasal], [+stop] (in whatever feature system you're using). Together these values uniquely identify the segment.
The power of the matrix representation is what it reveals about groups of sounds. Consider the set {p, b, t, d, k, g} — these are English stops. What do they all share? In feature notation, they all have the value [+stop] (or [−continuant] in some frameworks). That shared feature is the formal definition of their natural class: a group of sounds that share one or more features and that behave as a unit in phonological rules. Rules can then be written with feature specifications rather than lists of individual sounds. Instead of saying "voicing assimilation affects /p/, /t/, /k/," you say it affects [−voice, +stop], which is both more general and more explanatory.
Feature matrices earn their complexity in the description of phonological processes. When a rule changes /t/ to /d/ before a voiced consonant, the matrix representation makes the process transparent: only the value of [voice] changes from − to +; every other feature value stays the same. This partial-change representation is impossible with symbols alone. The matrix shows you *exactly* what changes and what remains constant — which is the linguist's goal when writing a rule.
The binary (±) convention assumes that most phonological oppositions are two-way: a sound is either voiced or voiceless, either nasal or oral, either labial or not labial. Some modern frameworks use multi-valued or privative features (where a feature is either present or absent, rather than + or −), but binary features remain the standard starting point because they map cleanly onto articulatory contrasts. As you move toward feature geometry, you'll see how these feature values are organized hierarchically — not as a flat list in a matrix, but as a tree where some features depend on others. The matrix is the entry point to that richer representational world.