Practice Makes Perfect Algebra - Carolyn Wheater [41]
This matrix tells us that 0x + 1y = 1, or simply y = 1. A few more row operations give us the complete solution.
This transformation of the matrix tells us that x = 5 and y = 1.
As we have seen, row operations can transform the augmented matrix associated with a system of equations to matrices that are equivalent; that is, they represent systems with identical solutions. This parallels the steps in an algebraic solution and therefore can be used to determine the solution of the system.
When row operations are used to transform the augmented matrix, the goal is to produce an equivalent matrix in which the solution is apparent. This form, in which the area originally occupied by the coefficients has been transformed to an identity matrix, is called reduced row echelon form.
To solve the system , we first create the augmented matrix . Many different sequences of row operations could be used to produce a reduced row echelon form matrix equivalent to this one. One possible series of transformations will be shown here.
This final transformation gives us a reduced row echelon form. Note the 3 × 3 identity matrix in the first three columns: . The final column contains the solution. To read the solution properly, consider a copy of the final matrix with the columns labeled with the variable whose coefficients it contains. Imagine an arrow from the variable to the 1 in the column below it. Make a 90° turn and follow across to the solution column as shown in Figure 13.9.
Figure 13.9 Reading the solution.
Using this method we read the value of each variable.
x = 7 y = −3 z = 4
Dependent and inconsistent systems
There are situations, as we know, when systems do not have unique solutions. Some systems are dependent, that is, one equation in the system is a constant multiple of another. Other systems may be inconsistent, meaning that two equations within the system contradict one another.
For an augmented matrix that represents a dependent system or an inconsistent system, the attempt to use the reduced row echelon form method will fail. The positions that ought to hold the identity matrix will not contain the proper values. The result may be close to an identity; this is why it is important to examine the result carefully.
For a dependent system the final matrix will contain a row composed entirely of 0s. The final matrix for an inconsistent system will contain a row composed of 0s in the coefficient positions and a non-0 value in the last column.
The system is dependent, because the equation 6x + 9y − 18z = 24 multiple of 2x + 3y − 6z = 8. The augmented matrix for this system is . When we put this matrix in reduced row echelon form, the result is . The third column does not contain the pattern of 1s and 0s that completes the identity, and the bottom row of the output is entirely 0s.
The system is inconsistent because the first and second equations are in obvious conflict. The augmented matrix for this system is . When we put this matrix in reduced row echelon form, we get . Again, the third column is incorrect, but the bottom row tries to tell us that 0x + 0y + 0z = 1, a clearly impossible result.
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EXERCISE 13.9
Transform each augmented matrix to reduced row echelon form.
Identify each system as consistent, inconsistent, or dependent.
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Answers
1 Arithmetic to algebra
1.1
1. Rationals, Reals
2. Integers, Rationals, Reals
3. Rationals, Reals
4. Whole, Integers, Rationals, Reals
5. Irrationals, Reals
6. Natural, Whole, Integers, Rationals, Reals
7. Rationals, Reals
8. Rationals, Reals
9. Rationals, Reals
10. Irrationals, Reals
1.2
1. Commutative Property for Addition
2. Associative Property for Multiplication
3. Identity for Addition
4. Inverse for Multiplication
5. Distributive Property
6. Zero Product Property
7. Associative Property for Addition
8. Identity for Multiplication
9. Commutative Property for Multiplication
10. Multiplication Property of Zero
11. Inverse for Addition
12. Distributive Property
13. Identity for Multiplication