A First Course in Linear Algebra, 2017a

1.2. Systems of Equations, Algebraic Procedures 15
Definition 1.10: Augmented Matrix of a Linear System
For a linear system of the form
a 11 x 1 + ···+ a 1n x n = b 1
...
a m1 x 1 + ···+ a mn x n = b m
where the x i are variables and the a ij and b i are constants, the augmented matrix of this system is
given by
⎡
⎤
a 11 ··· a 1n b 1
⎢
⎥
⎣
...
...
... ⎦
a m1 ··· a mn b m
Now, consider elementary operations in the context of the augmented matrix. The elementary operations
in Definition 1.6 can be used on the rows just as we used them on equations previously. Changes to
a system of equations as a result of an elementary operation are equivalent to changes in the augmented
matrix resulting from the corresponding row operation. Note that Theorem 1.8 implies that any elementary
row operations used on an augmented matrix will not change the solution to the corresponding system of
equations. We now formally define elementary row operations. These are the key tool we will use to find
solutions to systems of equations.
Definition 1.11: Elementary Row Operations
The elementary row operations (also known as row operations) consist of the following
1. Switch two rows.
2. Multiply a row by a nonzero number.
3. Replace a row by any multiple of another row added to it.
Recall how we solved Example 1.9. We can do the exact same steps as above, except now in the
context of an augmented matrix and using row operations. The augmented matrix of this system is
⎡
1 3 6
⎤
25
⎣ 2 7 14 58 ⎦
0 2 5 19
Thus the first step in solving the system given by 1.5 would be to take (−2) times the first row of the
augmented matrix and add it to the second row,
⎡
1 3 6
⎤
25
⎣ 0 1 2 8 ⎦
0 2 5 19