Algebra

A matrix is a rectangular array of numbers, and matrix algebra involves the formulation of rules for manipulating matrices. The elements of a matrix are contained in square brackets and named by row and then column. For example the matrix has two rows and two columns, with the element (-6) located in row one column two. In general, a matrix can have i rows and j columns, so that an element of a matrix is denoted in double subscript notation by a_ij. The four elements in A are a₁₁ = 1, a₁₂ = -6, a₂₁ = 3, a₂₂ = 2. A matrix having m rows and n columns is called an "m by n" or (m × n) matrix. When the number of rows equals the number of columns the matrix is said to be square. In matrix algebra, the operations of addition and multiplication are extended to matrices and the fundamental principles for combining three or more matrices are developed. For example, two matrices are added by adding their corresponding elements. Thus, two matrices must each have the same number of rows and columns in order to be compatible for addition. When two matrices are compatible for addition, both the associative and commutative principles of elementary algebra continue to hold. One of the many applications of matrix algebra is the solution of systems of linear equations. The coefficients of a set of simultaneous equations are written in the form of a matrix, and a formula (known as Cramer's rule) is applied which provides the solution to n equations in n unknowns. The method is very powerful, especially when there are hundreds of unknowns, and a computer is available.