A matrix is a concise way of representing and working with linear transformations; also, it is a rectangular array or grid of numbers or variables that allows the user to perform certain mathematical operations. They are usually symbolized as large parentheses or two large pair of parallel double lines surrounding the array of numbers or variables. These numbers can be manipulated to solve systems of equations or problems with many different variables or numbers, by addition, subtraction, multiplication or other methods. Each row and column of a given matrix must have the same number of elements.

Any time one has a list of numbers, or a table of numbers in a specific order, concerning anything at all (prices, grades, populations, coordinates of points, production tables …), it can be considered to be a matrix. When the idea of the matrix was first conceived, its development dealt with transformation of geometric objects and solution of systems of linear equations. Historically, the early emphasis was on the determinant (see below), not the matrix; today, especially in linear algebra, matrices are considered first.