Activity 2 - Useful Inverses
Day 3
Objectives:
- Relate the process of finding an inverse and methods of solving a system of equations.
- Examine the use of inverses in solving systems of equations.
- Write a system of equations as a matrix equation.
- Discover the relationships between inverses and determinants.
- Use technology to observe the process of finding an inverse.
- Discuss the methods that technology can use to calculate determinants and inverses.
Materials:
- Pencil and Paper
- Web applet: Online Matrix Calculator
- Web applet: MatrixCalc
In this handout, answers are in orange and suggestions/directions are in red.
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Description of Activity:
In this activity, students examine the use of matrices in solving systems of equations. Students perform multiplication using the coefficient and variable matrices to confirm that the two representations of the system of equations are equivalent. They then use the inverse of the coefficient matrix to solve the system of equations. Finally, the teacher demonstrates how to find an inverse without the use of technology. The second web applet, MatrixCalc, is used so that students can observe that the calculator performs the same steps.
After observing inverses, students examine determinants and how they change under the inverse map. The teacher then demonstrates how to calculate the determinant. MatrixCalc is again used so that students can observe how the applet calculates the determinant.
Rationale:
This activity covers the standard MCC9-12.A.REI.9, which includes using technology to calculate inverses. The web applets are not only used to hasten calculations, but also to demonstrate the shortcomings of technology. Specifically, students must do #1 by hand since the Online Matrix Calculator cannot handle variable entries. This can lead to a meaningful discussion on the usefulness of technology and the need for the students to know the processes even if the work can usually be done for them. Again, the web applets allow students to work with inverses before learning how to calculate them, which can make them more motivated to learn the algorithms. A similar phenomenon should occur with determinants as well. Using the MatrixCalc applet allows students to observe how the technology does their work. It is valuable to see that it will calculate inverses in a similar fashion, but use a trick to calculate the determinant more quickly. These agreements and discrepancies in strategies can allow for in-depth conversations at a high level of understanding.
In this activity, students examine the use of matrices in solving systems of equations. Students perform multiplication using the coefficient and variable matrices to confirm that the two representations of the system of equations are equivalent. They then use the inverse of the coefficient matrix to solve the system of equations. Finally, the teacher demonstrates how to find an inverse without the use of technology. The second web applet, MatrixCalc, is used so that students can observe that the calculator performs the same steps.
After observing inverses, students examine determinants and how they change under the inverse map. The teacher then demonstrates how to calculate the determinant. MatrixCalc is again used so that students can observe how the applet calculates the determinant.
Rationale:
This activity covers the standard MCC9-12.A.REI.9, which includes using technology to calculate inverses. The web applets are not only used to hasten calculations, but also to demonstrate the shortcomings of technology. Specifically, students must do #1 by hand since the Online Matrix Calculator cannot handle variable entries. This can lead to a meaningful discussion on the usefulness of technology and the need for the students to know the processes even if the work can usually be done for them. Again, the web applets allow students to work with inverses before learning how to calculate them, which can make them more motivated to learn the algorithms. A similar phenomenon should occur with determinants as well. Using the MatrixCalc applet allows students to observe how the technology does their work. It is valuable to see that it will calculate inverses in a similar fashion, but use a trick to calculate the determinant more quickly. These agreements and discrepancies in strategies can allow for in-depth conversations at a high level of understanding.
Standards Addressed By This Activity:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
- MCC9-12.N.VM.8 - Add, subtract, and multiply matrices of appropriate dimensions.
- MCC9-12.N.VM.9 - Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- MCC9-12.N.VM.10 - Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- MCC9-12.A.REI.8 - Represent a system of linear equations as a single matrix equation in a vector variable.
- MCC9-12.A.REI.9 - Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater).
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure.