Ideas from Classroom Teachers for
Systems of Linear Equations and Matrices
Modeling and solving problems using matrices should be integrated throughout this unit.
Use of a graphing calculator after doing pencil/paper work would be appropriate here.
I place students in pairs and have them do some research on matrices and their use in solving problems. The students find a specific problem to solve using matrices, present it to the class, and write a 2-page paper with cover page and list of sources. This is done in conjunction with English teachers, who help with formatting. These same students do a major research paper at a later date for their English class. The mathematics paper serves as practice. Some of the problems discussed include dominance matrix, communications matrix, and using Cramer’‘s Rule to solve systems of linear equations. A helpful website is http://en.wikipedia.org/wiki/matrix_(mathematics).
Helpful book: Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory by Dennis Bernstein.
Encourage students to check their solutions by substituting back into the original equations.
Technology note: Graphing calculators could be used to find solutions, then students can verify these solutions algebraically.
Linear systems may have one solution, no solution, or infinitely many solutions. Help students recognize each case both algebraically and graphically.
Historical note: Matrix algebra in its modern sense was developed by Leibniz, Cauchy, Gauss (and others) in the 18th century. It is a relatively new concept in mathematics history.
Students can discover through examples that matrix multiplication is not commutative.
Technology note: Calculators can enhance students’‘ understanding of matrices by quickly performing matrix operations and finding inverses and determinants.
Application suggestion: Matrices can be used to represent points in the coordinate plane and to perform transformations (e.g., reflections, rotations, scale changes, etc.) on points.
Before matrix addition, I would spend a day or so on how to organize numerical information in matrices.
I also suggest one day on inverses of matrices (what happens when you multiply a matrix by its inverse, etc.).
It is extremely important that students get a feel for how we can get no solution or infinitely many solutions (and what exactly this means for an application problem).
The problems that are modeled can get pretty involved. It is conceivable that students will only be able to get through one or two good problems in a class period.
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