Operations with Matrices (2 of 2)
Lesson 3 of 10
Objective: SWBAT multiply matrices.
Today we continue our work with matrix operations by focusing on how to multiply matrices. Students start with Bell Work that reminds them of the concepts learned yesterday. In particular, it focuses on you can evaluate expressions involving matrices.
I give students a few minutes to work on the questions. After students determine which expressions can be found I ask two general questions:
- What operation can be done no matter what the dimensions of the matrix?
- What is true to be able to add and subtract matrices?
I hope that my students respond that scalar multiplication can always be performed, but addition and subtraction require the matrices to have equal dimensions.
Matrix multiplication can be hard for students to understand, but using a contextual problem helps students appreciate matrix multiplication. To get things started, I give students the How to Multiply Matrices activity. This activity develops the process of multiplying matrices. Students multiply without writing a matrix, then they are given the matrices that could be used to find the answer they found. I give students about 10 minutes to work on the activity.
After ten minutes, we will focus on questions 3 and 4 since these questions put things together. I ask students to explain how the matrix multiplication is performed. We discuss how we take the row of the first matrix and multiply each term in a column of the second matrix then add the product to get a result. I ask the class, "What would happen if we added a 4th brand to the problem?"
Then, I have students share the matrix they wrote for question 4. I expect that some students will make a column matrix and others will make a row matrix. It is important to ask students why they wrote the answers as a row or column matrix. As we discuss this question, I will mention that we are multiplying Row 1 of the first matrix and Column 1 (or 2 or 3) of the second matrix. I ask, "What makes sense for recording our results? What order matrix?" We will continue to review answers until the students agree the answer matrix should be a 1X3 matrix.
I now want students to practice some multiplication. I give students problems. The first 2 problems give students a quick practice. Students use paper or whiteboards to do the problems. Answers are shared and I discuss the dimensions of the matrices.
The third problem is not possible. I allow students to work for a time. Once student are not sure what to do I ask the class:
- Is there something wrong with this problem?
- Why can't you find the answer?
- What has to be true to multiply matrices?
- What will the result dimensions be?
I now comment that multiplication can be tedious when the matrices become large. Our calculators can do matrix operations. I give students directions for Matrices on the TI-84. I have the students use the calculator to verify our answers on the practice problems. I ask students about using the calculator.
- Is the calculator make doing operations on matrices quicker? easier?
- If you are taking a timed test and you have a 2X2 multiplied with a 2X2 would you use the calculator?
As today's class ends I have the students get out the Summary of Operations worksheet from yesterday's closure. I have students add multiplication of matrices and write out how a description of how to complete the operation for the students use.
I also assign some problems for the students to complete.
From Larson Precalculus with Limits, 2nd ed. p. 595 #29, 32, 33, 39, 40, 48.
These problems begin by reviewing the operations from the previous day and then work with multiplication. The students determine whether they can find the product. The last problem gives students several products to determine if they are possible. If possible the students find the product.