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A matrix is a rectangular array of numbers (or other mathematical objects) arranged in rows and columns, whose entries are typically combined using standard operations such as addition and multiplication. An m×n matrix has m rows and n columns, and its entries are often denoted by two subscripts indicating the row and column position (for example, a2,1 is the element in row 2, column 1). Matrices are central in many areas of mathematics and science, especially in linear algebra, where they represent linear maps and are used to describe geometric transformations and coordinate changes. More formally, a matrix is defined as a rectangular array of entries, usually taken from a field (or sometimes a ring). Matrices support operations like addition (performed entrywise), scalar multiplication (multiplying each entry by a scalar), transpose (swapping rows and columns), and matrix multiplication (corresponding to composition of linear transformations). Square matrices (same number of rows and columns) are particularly important; for them, concepts such as determinant, invertibility, and eigenvalues/eigenvectors play a major role in understanding the matrix’s properties.
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The size (or dimensions) of a matrix is determined by the number of rows and the number of columns it contains. A matrix with m rows and n columns is called an m × n matrix (or m-by-n matrix), where m and n are its dimensions. Matrices with one row are row matrices (row vectors), and matrices with one column are column matrices (column vectors). A matrix with the same number of rows and columns (m = n) is a square matrix, which is especially important in matrix theory. Matrix dimensions also extend to special cases: an infinite matrix has infinitely many rows and/or columns, and an empty matrix has no rows or no columns (useful in some contexts like computer algebra). The entries of an m × n matrix are typically indexed by row and column numbers (i, j), and the matrix is often written using bracket or parenthesis notation. Common conventions denote the matrix by an uppercase letter (e.g., A) and its entries by lowercase letters with two subscripts (e.g., a_ij).
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