How To Teach with DWFK Precalculus

Chapter 7: Systems and Matrices

Matrices do not appear in most first-year calculus courses, so over the years they have gotten less than their proper share of attention in precalculus classes. Ironically, it is their importance for all the other applications of mathematics that makes them so important today. We recommend, therefore, that all classes work through at least the first three sections of this chapter. Students who go on to calculus will need matrices in the second year, and all students are likely to see matrices in some form sooner than that.

Section 7.1 Section 7.3 Section 7.5
Section 7.2 Section 7.4

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Section 7.1 Solving Systems of Two Equations

Objectives
Students will be able to solve systems of equations graphically and algebraically.

Key Ideas
Demand curve Solution by substitution
Equilibrium point Solution of a system
Equilibrium price Supply curve
Solution by elimination System of equations

Teaching Tips
Many students will have already seen the substitution and elimination methods for solving systems of linear equations, but a light review does not hurt. Be sure to emphasize the graphical interpretation of finding intersection points of lines. Not only does this clarify the phenomena of empty solutions and infinite solutions, but it serves as an easy bridge to the graphical solution of non-linear systems. Algebraic solutions of non-linear systems can be quite difficult in general, so our non-linear, algebraic examples are deliberately chosen to be simple.

Technology Tips
Calculators that will solve linear systems can obscure the algebraic theory that this section is intended to highlight, so explain to students that we are less interested (for now) in knowing they can get the answers than we are in knowing that they understand these pencil-and-paper methods. Assure them that we will all get technological in Section 7.3.
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Section 7.2 Matrix Algebra

Objectives
Students will be able to find sums, differences, scalar multiples, and products of matrices. They will be able to find the inverses of matrices by hand and the inverses of larger square matrices by using a calculator.

Key Ideas
Additive inverse Non-singular matrix
Cofactor Order of a matrix
Column Row
Determinant Scalar
Element (entry) of a matrix Singular matrix
Inverse of a matrix Square matrix
Matrix Transpose of a matrix
Minor Zero matrix

Teaching Tips
Teachers might well decide to introduce this material much earlier in the course, since it is not dependent on anything in the first six chapters. Also, there are advantages to having matrices available for examples in other sections. For example, it is an interesting bit of trigonometry to prove that

Be sure that students understand the application of matrix multiplication in Example 5. (There are several exercises that test this understanding.) Computer spreadsheets do matrix multiplications in the blink of an eye, but it is still up to computer users to set up the multiplications properly.

The fact that matrix multiplication is not commutative is a significant algebraic discovery for students and should not be passed over lightly.

The general determinant algorithm of expansion by cofactors, given on page 559, is shown in this section for the sake of completeness. It is not recommended that students find determinants by hand for matrices larger than , unless they know enough about row-reduction to make the task reasonable. (For example, students who compete in mathematics contests might be challenged to find larger determinants, but the secret is usually to reduce them first.)

There is a fairly well-known trick for computing determinants, but teaching it to students might do more harm than good since it does not generalize at all to matrices of other orders. If the object is to understand determinants, teach cofactors. If the object is to find determinants, use a calculator. In this course, the object is usually to find determinants (or, more often, to find inverse matrices).

Technology Tips
Feel free to use the calculator quite freely in this section. Matrices were invented to take some of the tedium out of computations, and if the calculator can make things even less tedious, then they are right in step with the program.

The section has quite a few technology tips in it already, but here is one more. Not many calculator users realize that they can build matrices on the home screen by using square brackets. For example, [[1,2,3][4,5,6][7,8,9]] is read as a matrix. The following screen shows how to type it on the home screen and store it as matrix [A]:


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Section 7.3 Multivariate Systems and Row Operations

Objectives
Students will be able to solve systems of linear equations using Gaussian elimination, the reduced row echelon form of a matrix, or matrix inversion.

Key Ideas
Augmented matrix Invertible square linear system
Coefficient matrix Reduced row echelon form
Equivalent linear systems Row echelon form
Gaussian elimination Triangular form

Teaching Tips
The material in this section is not necessary for first-year calculus, but it explains the mathematics behind two time-saving calculator approaches to solving systems of linear equations. Since students will find the technique of Examples 7, 8, and 9 to be the easiest, teachers who are pressed for time might want to consider omitting the other examples. Note, however, that the solution only works on invertible square linear systems. Example 3 (no solution) and Examples 5 and 6 (infinitely many solutions) are best solved by Gaussian elimination or by interpreting the reduced row echelon form of the augmented matrix. Note that the calculator can be used to find reduced row echelon form.

Technology Tips
The solving of simultaneous linear equations is important for many real-world problems in elementary algebra, and most of them are modeled by invertible square linear systems. Since the modeling in this case is more important than the manipulations required to solve the systems, the day is not far off when Algebra I students will be solving such problems using (or possibly some user-friendly menu item) on their graphing calculators. The slower and riskier "pencil-and-paper" methods should therefore be studied for their historical and pedagogical value, not because they will actually be used.
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Section 7.4 Partial Fractions

Objectives
Students will be able to decompose certain rational expressions into partial fractions.

Key Ideas
Partial fractions
Partial fraction decomposition

Teaching Tips
This little nugget of algebra is useful for finding antiderivatives of rational expressions in first-year calculus, although many calculus teachers skip those antiderivatives precisely because students do not know how to find the partial fractions. Realistically, it is not necessary for students to learn this process well. Those who have calculus teachers who like the topic will have it explained to them again, and those who have calculus teachers who skip the topic might never see it at all.

The AB Calculus course description for AP Calculus does not include partial fractions. The topic is in the BC course description, but students only need to be able to decompose rational expressions with unrepeated linear factors in the denominator (as in Example 1).

Technology Tips
Graphing calculators with computer algebra systems (CAS) will decompose rational expressions into partial fractions. Indeed, they will find the solutions to the differential equations that partial fractions are needed to solve. These two facts suggest that it is only a matter of time before this topic disappears from calculus and precalculus courses entirely. Meanwhile, if you want your students to learn the algebraic manipulations required to find partial fractions, be sure to restrict the use of CAS calculators.
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Section 7.5 Systems of Inequalities in Two Variables

Objectives
Students will be able to solve linear programming problems and systems of inequalities using graphical methods.

Key Ideas
Constraints Objective function
Half-plane System of inequalities
Linear programming problem Vertex (corner) points of a feasible region

Teaching Tips
Linear inequalities are covered in this section so that linear programming can be included in the textbook. (Linear programming is a syllabus item on some state requirement lists.) As far as preparation for calculus is concerned, this section is an interesting detour.

Notice the shading of the region in Figure 7.32. Shading only the region that satisfies all the constraints is vastly preferable to shading the constraints one at a time and searching the cross-hatched diagram for the intersection of all the shadings.

Technology Tips
Shown below is the way to select the shading "styles" necessary to shade the inequalities in Example 7. The result, as you can see, is too muddy to be very useful.

A clever way to get a readable graph is to shade everything in the opposite direction! The feasible region that satisfies all the constraints then shows up in white, as shown below.