We return to the problem touched upon in the previous section, and discuss

the Gaussian elimination method for solving the following system of n linear

simultaneous equations in the n unknowns x1, x2, . . . , xn

a11x1a12x2 a1nxnk1 a21x1a22x2 a2nxnk2

an1x1an2x2 annxnkn: Such a system is said to be non-homogeneous (inhomogeneous) if not all of

the k1, k2, . . . , kn are zero, and homogeneous if k1/k2/. . ./kn/0. The system may be written in the matrix form




a11 a12 a1n a21 a22 a2n

an1 an2 ann

2 664

3 775, X

x1 x2 n


2 664

3 775 and B

k1 k2 n


2 664

3 775:

For obvious reasons, the matrix A is called the coefficient matrix. If A is non-singular, an inverse matrix A1 exists, so premultiplying the

matrix equation by A1 we have


but A1A/I and IX/X , so this reduces to


Thus, when A is non-singular the system has a unique solution and, in particular, if B/0 so that A1X/0, the unique solution becomes X/0, usually called the trivial solution. If, however, matrix A is singular, there will 1 fail, a