Do Matrices Work Like Real Numbers?

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Objective

SWBAT identify which real number properties can be used with matrices.

Big Idea

Through exploration students discover that some properties of real numbers will not exist for matrices.

Bell Work

5 minutes

For today's Bell Work I ask students to determine whether the given pairs of matrices can be added and multiplied. I will have my students discuss this in groups for a couple of minutes. By doing this work students review the matrix operations. I expect that some students will get out the Summary of operations worksheet from the last 2 lessons to help them remember how to multiply. Then, as a class we discuss each example and students explain their conclusions. A few students, at first, think the second example can be multiplied. I have students in the class explain why they think this is possible, why they think that this is not possible.  

Creating matrices to fit a situation

10 minutes

I now give student two open questions to work on with a partner. These 2 questions help students see that we sometimes have matrices that we can add but not multiply or vice-versa. I let students work for several minutes to make the matrices. When a couple of pairs have determined their matrices, I ask pairs to start forming groups to check each others results. I will do this until all pairs have shared with another pair.

Afterward, I will ask some groups to share sets of matrices on the board and so that we can discuss some different types of examples as a class. I ask the students who share to explain their reasoning when making their matrices. I find that hearing how other students think about a process like this helps students develop productive strategies and build conceptual understanding of them.

Discovering Properties of Matrices

20 minutes

After the previous activity, students understand that some operations that always work on real numbers will not always be possible with matrices. I let them know that this applies to some properties they have learned for real numbers.

Our next activity will help students discover which properties are true for matrices. The task is included in the resource, Matrices Properties. For this activity, I ask students to work in groups and I give students directions for using the calculator to perform matrix operations. Being able to input the matrices by their name and then doing the calculations will help students complete this activity in an efficient amount of time. In this case, using the time efficiently will help students focus on the properties, not on computation.

As students work I move around the room and ask questions such as:

  • Why did the commutative property not work for matrices?
  • Can we sometimes switch the matrices around and get a result? Will the result be the same?

After students have worked for about 15 minutes on Matrices Properties the class comes back together to discuss what they have discovered. We can typically cover the first three properties quickly. The identity properties provide a little more resistance.

When we discuss the additive identity, I ask my students, "What does the identity matrix looks like?" Students will generally say, "All the elements are zero, and, it has the same dimension of the original matrix." I respond by saying, "What do you think mathematicians call a matrix that has all zeros?" Students often assume that the obvious answer works. They'll say, "zero matrix." I'll respond by saying, "Right, mathematicians are really creative with names aren't they?"

When we get to the multiplicative inverse students need more guidance on how to determine the type of matrices that have inverses and how to make the identity matrix. Some guiding questions include:

  • To verify the inverse we need to be able to switch the matrices and still get a product. What types of matrices can we switch and get a product?
  • What do you notice about the 2X2 identity matrix. How do you think this can be expanded to get the identity matrix for a 3X3 or a 4X4?

By the end of this discussion students are typically beginning to see that you need a square matrix with the major diagonal all "1" and the remaining elements all "0". It is accepted, but perhaps not yet a comfortable idea to work with for most of my students. 

Closure

5 minutes

As class ends students answer a summary question as an Exit Slip. We are at a point where I want students to be specific about dimension with respect to different operations with matrices.  I am interested in other things that students may mention. This brief summary gives me the opportunity to see where students are in their understanding of matrix operations.