Page 57 - A Course in Linear Algebra with Applications

2.2: Elementary Row Operations 41

6. For which values of t is the following linear system consis-
tent?

7. How many operations of types (a), (b), (c) are needed in
general to put a system of n linear equations in n unknowns
in echelon form?

2.2 Elementary Row Operations

If we examine more closely the process of Gaussian elim-
ination described in 2.1, it is apparent that much time and
trouble could be saved by working directly with the augmented
matrix of the linear system and applying certain operations
to its rows. In this way we avoid having to write out the
unknowns repeatedly.
The row operations referred to correspond to the three
types of operation that may be applied to a linear system dur-
ing Gaussian elimination. These are the so-called elementary
row operations and they can be applied to any matrix. The
row operations together with their symbolic representations
are as follows:
(a) interchange rows i and j , (i?j <-»• Rj);
(b) add c times row j to row i where c is any scalar,
(Ri + cRj);
(c) multiply row i by a non-zero scalar c, (cRi).

From the matrix point of view the essential content of The-
orem 2.1.2 is that any matrix can be put in what is called
row echelon form by application of a suitable finite sequence
of elementary row operations. A matrix in row echelon form