Book: A Memoir on the Theory of Matrices

Overview
Arthur Cayley lays out a systematic algebra of matrices that transforms arrays of numbers from mere shorthand into genuine algebraic objects. He treats matrices as entities subject to addition and multiplication, explores their relation to determinants, and uses them to represent linear substitutions and systems of linear equations. The exposition establishes foundational rules and demonstrates how matrix operations encode composition of linear transformations, anticipating much of modern linear algebra.

Definitions and Notation
Cayley introduces a clear notation for matrices and their elements, distinguishing square matrices from rectangular arrays and identifying special cases such as diagonal and unit matrices. He defines addition entrywise and frames multiplication so that the product corresponds to successive application of linear maps; this definition is motivated by operation on column vectors. Attention is paid to index notation and to conventions that permit concise manipulation of symbols representing matrix entries.

Algebraic Structure and Operations
The memoir develops the basic algebraic laws that govern matrices. Cayley demonstrates associativity of multiplication and explains why multiplication need not be commutative, giving explicit examples to show that AB and BA generally differ. Rules for distributivity and scalar multiplication are given, and explicit formulas are worked out for the products of small matrices as illustrations. These formal properties are used to create calculational techniques that parallel those of ordinary algebra while reflecting the novel noncommutative nature of matrix multiplication.

Determinants, Inverses and the Characteristic Polynomial
Determinants receive systematic treatment as functions associated with square matrices, and Cayley explores their role in determining invertibility. The adjugate (classical adjoint) and its relation to the inverse of a matrix are explicated through determinant cofactors, providing practical methods for solving linear systems. A central highlight is the assertion that a matrix satisfies its own characteristic polynomial, an idea that later becomes known as the Cayley–Hamilton theorem. Cayley shows how the characteristic equation connects eigenvalues, determinants, and traces and uses it to derive identities that simplify computations and structural reasoning about matrices.

Applications to Linear Equations and Transformations
Matrices are applied directly to linear systems, with the algebraic framework used to express systems compactly and to manipulate them via matrix inversion and elimination-like procedures. Cayley interprets matrices as representations of linear substitutions acting on vectors and on coordinates in geometry; composition of substitutions corresponds to matrix multiplication. He illustrates how algebraic relations among matrices reflect geometric and transformational properties, providing examples that link the abstract operations to classical problems in invariant theory and the theory of forms.

Examples, Special Cases, and Techniques
Concrete examples for 2×2 and 3×3 cases are worked out in detail, showing explicit computation of products, determinants, and inverses, and illuminating exceptional behaviors that arise in noncommutative settings. Special matrix polynomials and simple canonical forms are considered as a way to simplify repeated operations, and methods for reducing matrices to simpler representatives under similarity transformations are sketched. Computational rules for minors and cofactors are collected to aid practical calculations.

Legacy and Influence
The memoir establishes many of the basic concepts that become central to linear algebra and matrix theory. By treating matrices as algebraic objects with their own operations and identities, Cayley creates a framework that unifies treatments of systems of equations, linear transformations, and determinants. Subsequent developments in matrix theory, invariant theory, and applications across geometry and mathematical physics trace roots to the ideas and techniques set out here, which moved matrices from computational devices to core mathematical structures.