ABSTRACT
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
AXB,
where
A
a11 a12 a1n a21 a22 a2n
an1 an2 ann
2 664
3 775, X
x1 x2 n
xn
2 664
3 775 and B
k1 k2 n
kn
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
A1AXA1B,
but A1A/I and IX/X , so this reduces to
XA1B:
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
tion.
