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8.1
Matrices and Systems of Equations
59 1
A matrix derived from a system of linear equations (each written in standard
form with the constant term on the right) is the augmented matrix of the
system. Moreover, the matrix derived from the coefficients of the system (but
not including the constant terms) is the coefficient matrix of the system.
System
Augmented Matrix
x - 4>· + 3z = 5
-x + 3y- 2,= -3
2x
- 4z = 6
Coefficient
-4
3
3 -1
0 - 4
1
-1
5
-3
6
2
Matrix
-4
3
0
3
-1
-
4
Note Note the use of 0 for the missing v-variable in the third equation, and
also note the fourth column (of constant terms) in the augmented matrix.
When forming either the coefficient matrix or the augmented matrix of a
system, you should begin by vertically aligning the variables in the equations
and using O's for the missing variables.
Given System
x + 3y =
Line Up Variables.
9
x +
->' + 4z = -2
x- 5z=
Q
=
3y
Form Augmented Matrix.
9
-y + 4z = -2
x
- 5z = 0
3
0
9
0 - 1
1
4
-2
1 0 - 5
0
Elementary Ro w Operation s
In Section 7.3, you studied three operations that can be used on a system of
linear equations to produce an equivalent system.
1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.
In matrix terminology, these three operations correspond to elementary row
operations. An elementary row operation on an augmented matrix of a given
system of linear equations produces a new augmented matrix corresponding to
a new (but equivalent) system of linear equations. Two matrices are rowequivalent if one can be obtained from the other by a sequence of elementary
row operations.
Elementary Row Operations
1. Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.
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