Page 58 - A Course in Linear Algebra with Applications

42 Chapter Two: Systems of Linear Equations

has the typical "descending staircase" form

0 ll * • * * * • • * * • * * \
0 0 0 0 • • 0 1 * • • * * • * *

0 0 0 0 • • 0 0 0 •• ll * • * *
0 0 0 0 • • 0 0 0 •• 0 0 • • 0 *

Vo 0 0 0 • • 0 0 0 •• 0 0 • • 0 */

Here the asterisks denote certain scalars.

Example 2.2.1

Put the following matrix in row echelon form by applying
suitable elementary row operations:

1 3 3 2 1
2 6 9 5 5
- 1 - 3 3 0 5

Applying the row operations R2 — 2R\ and -R3 + R\, we
obtain
1 3 3 2 l \
0 0 3 1 3 .
0 0 6 2 6 /

Then, after applying the operations |i?2 and R3 — 6R2, we
get
1 3 3 2 1
0 0 1 1 / 3 1
0 0 0 0 0
which is in row echelon form.

Suppose now that we wish to solve the linear system with
matrix form AX = B, using elementary row operations. The
first step is to identify the augmented matrix M = [A \ B].