Enter a tone row and get the whole matrix — all forty-eight forms of the row in one table, each one labelled. Read across for prime, back for retrograde, down for inversion, up for retrograde inversion.
Twelve-tone composition starts from a row: an ordering of all twelve pitch classes with none repeated until the rest have been used. The row is not a melody — it is a source of material, and it is used in four forms. The original ordering is the prime (P), the same ordering backwards is the retrograde (R), the same intervals in the opposite direction is the inversion (I), and the inversion backwards is the retrograde inversion (RI). Each of those can start on any of the twelve pitch classes, which gives forty-eight forms in total.
Writing forty-eight rows out by hand is tedious and error-prone, so composers use a single 12×12 table that contains all of them. Read a line left to right and you have a prime form; read the same line right to left and you have its retrograde. Read a column downwards for an inversion, upwards for a retrograde inversion. The number in each label is the pitch class the form starts on, so P0 begins on 0 and I5 begins on 5.
The row is first transposed so that it begins on 0 — that normalised row is the top line. The left-hand column is the inversion of that row, which is simply each interval flipped in direction. Every other cell is then the top-row value transposed by its left-column value. The diagonal is all zeros, which is a quick way of checking the table is right.
A matrix tells you which forms share the most content, which is how serial composers choose the next row form: look for a P and an I whose first few notes overlap, or a hexachord that appears in both, and the join between them will sound connected rather than arbitrary. That is the difference between using the system and being used by it.
Serialism, what it was reacting against and what came after it are covered in the Cloud Atelier guides; the Music Composition and Counterpoint & Advanced Harmony paths put it in order. For the tonal harmony this music deliberately left behind, start with the Circle of Fifths and the Chord Progression Analyzer.