Inverses and Determinants
Lesson 5 of 10
Objective: SWBAT compute determinants and use them to find the inverse of a matrix
I begin today by reminding students about the concept of an inverse. For example, we have already discussed inverses as when you take the composition of a function and its inverse your result will be x. Today, we will discuss the inverse of a number and how the result gives the identity of 1.
Students begin their work today with a Bell Work Question. After students have reflected on the question, I have several students share what they think. I then put up numbers for students to find the multiplicative inverse. At first students may give the opposite. We discuss how this is the additive inverse. I will ask "why is it the additive inverse?" Here I explain that 0 is the additive identity. If you take a number add the identity to the number you get that number back. So, for multiplication we need to think of a number when we multiply it with the original number we get the multiplicative identity. I'll ask, "What is the multiplicative identity?"
Once the class realizes that "1" is the multiplicative identity, we find the inverses and move to remember what the multiplicative identity was for matrix multiplication. I explain that today we will find the multiplicative inverse of a matrix. Finding the inverse of a matrix will help use solve problems involving matrices.
Now that students understand we are developing a method for finding the inverse of a matrix, I provide students with our book's brief introduction to the determinant. Our text goes into the detail of how it comes from solving a system of equations, but at this point I will omit that information.
Next, I plan to give students several matrices with the Determinant already calculated. I ask students to look for a pattern and discuss how the Determinant is calculated. As students work I will circulate to determine who has a productive contribution to make to a class discussion. Once most groups have discovered a pattern, I will ask a student (or two) to explain how to find a determinant for a 2X2 matrix.
I want to make sure my students understand the process, as explained, so I will put 3 problems on the board for students calculate the Determinant. After students have found the Determinant we share the process and the answer on the board. I next put up the full book definition (p. 2) of a determinant for a 2X2 matrix. I explain that in class we will focus on the 2X2 matrices, but the textbook explains how to find the Determinant for other square matrices.
I let students know that we will see several uses for determinants, the first is finding the multiplicative inverse. I give students a rule for finding the inverse of matrix by using the Determinant. We discuss what the rule is stating especially what is being stated in the matrix of the rule. I am aware and pay attention to the fact that understanding how the signs change on the major diagonal and how the terms switch on the minor diagonal is sometimes confusing, even when looking at the notation.
Next, I ask students to find the inverse of a matrix. To verify the inverse, we will use graphing calculators. This can be done in 2 different ways. The first is to put the original matrix into the calculator and use the inverse key. The other is to multiply the original matrix with the inverse matrix that was found. I prefer the second method since students see how multiplying the original equation with its inverse results in the identity matrix.
I now post a statement that is found in some books and ask the students to think about how this can be the case.
If the Determinant is 0, then the matrix does not have an inverse.
I let students discuss this hypothesis in their groups and we then share out the comments. I ask them to think of a 2X2 matrix that will not have an inverse. Oftentimes, an immediate response is the zero matrix. I am ready for this and I will say, "Okay, let's create another 2X2 matrix that does not have an inverse." It will take students several minutes to complete this problem. I'll observe as they work and make a decision as to a good point to stop and have students share their results.
As class ends, I plan to ask students to write an explanation to a classmate about how to find the determinant of a 2x2 matrix. I will ask my students to turn this task in as they leave class.
I also give students practice problems on determinants and inverses. These problems are only 2X2 matrices. I will show students tomorrow how to find the determinant using their calculator so we can use determinants to solve problems.