Page 13 - Matrix Analysis & Applied Linear Algebra

1.2 Gaussian Elimination and Matrices 5

Providing explanations for why each of these operations cannot change the
solution set is left as an exercise.
The most common problem encountered in practice is the one in which there
are n equations as well as n unknowns—called a square system—for which
there is a unique solution. Since Gaussian elimination is straightforward for this
case, we begin here and later discuss the other possibilities. What follows is a
detailed description of Gaussian elimination as applied to the following simple
(but typical) square system:
2x + y + z = 1,
(1.2.4)
6x +2y + z = − 1,
−2x +2y + z = 7.

At each step, the strategy is to focus on one position, called the pivot po-
sition, and to eliminate all terms below this position using the three elementary
operations. The coefficient in the pivot position is called a pivotal element (or
simply a pivot), while the equation in which the pivot lies is referred to as the
pivotal equation. Only nonzero numbers are allowed to be pivots. If a coef-
ficient in a pivot position is ever 0, then the pivotal equation is interchanged
with an equation below the pivotal equation to produce a nonzero pivot. (This is
always possible for square systems possessing a unique solution.) Unless it is 0,
the first coefficient of the first equation is taken as the first pivot. For example,
the circled in the system below is the pivot for the first step:
2
x + y + z = 1,
2
6x +2y + z = − 1,
−2x +2y + z = 7.
Step 1. Eliminate all terms below the first pivot.

• Subtract three times the first equation from the second so as to produce the
equivalent system:
x + y + z = 1,
2
− y − 2z = − 4 (E 2 − 3E 1 ),
−2x +2y + z = 7.

• Add the first equation to the third equation to produce the equivalent system:
x + y + z = 1,
2
− y − 2z = − 4,
3y +2z = 8 (E 3 + E 1 ).