In linear algebra, the coefficient matrix refers to a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
In general, a system with m linear equations and n unknowns can be written as
a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋮ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = b m where x 1 , x 2 , . . . , x n are the unknowns and the numbers a 11 , a 12 , . . . , a m n are the coefficients of the system. The coefficient matrix is the mxn matrix with the coefficient a i j as the (i,j)-th entry:
[ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] 