A matrix is a rectangle that eats vectors and spits maps
Stack numbers in rows and columns and you can add them, multiply them, and—when the array is square—ask about determinants and eigenvalues. That humble rectangle became the workhorse of linear maps, geometric transforms, and large-scale numerics.
A matrix is a rectangular array of entries—numbers or other objects—arranged in rows and columns. An m × n matrix has m rows and n columns; there is no fixed upper bound on those positive integers. Single-row and single-column matrices are row and column vectors; equal dimensions make a square matrix; infinite or empty matrices appear in specialized settings. Notation usually uses brackets or parentheses, capital letters for the whole array, and double subscripts for entries. Real and complex matrices are named for their entry fields.
In linear algebra matrices represent linear maps once bases are chosen. Geometry uses them for rotations and coordinate changes. Numerical analysis reduces many computations to matrix arithmetic, sometimes at enormous scale. Square matrices dominate theory: a nonzero determinant marks invertibility, and eigenvalues are roots of the characteristic polynomial. Matrix theory began as a corner of linear algebra and grew into graph theory, combinatorics, statistics, and algebra more broadly.
Operations obey size rules: you add matrices of the same shape entrywise; you multiply an m × n matrix by an n × p matrix to get an m × p result. That multiplication encodes composition of linear maps and is generally noncommutative—one reason matrices supply classic examples of noncommutative rings. From a 2 × 3 teaching example to sparse arrays in scientific computing, the rectangle remains the standard packing of linear data. Learn the size rules once, and matrix arithmetic unlocks half of applied mathematics.
Source: Matrix (mathematics)
