A set of m linear equations in n unknowns x1, x2,…, xn that has the form
a11x1 + a12x2 + ⋯ + a1nxn = b1,
a21x1 + a22x2 + ⋯ + a2nxn = b2,
am1x1 + am2x2 + ⋯ + amnxn = bm,
where b1, b2,…, bm are not all zero. (Compare with homogeneous set of linear equations.) In matrix notation, this set of equations may be written as Ax=b, where A is the m×n matrix [aij], and b(≠0) and x are column matrices:
Such a set of equations may be inconsistent, have a unique solution, or have infinitely many solutions (see simultaneous linear equations). For a set consisting of the same number of linear equations as unknowns, the matrix of coefficients A is a square matrix, and the set of equations has a unique solution, namely x=A−1b, if and only if A is invertible.