# Engaging students: Finding the inverse of a matrix

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Donna House. Her topic, from Algebra: finding the inverse of a matrix.

How could you as a teacher create an activity or project that involves your topic?

Engage the students by asking them how they think our military (or a secret agent) sends and receives messages without the enemy knowing what message is being sent. Then the discussion can be guided by asking how math is used in encoding and de-coding secret messages. Since they already will have learned about matrices, tell them they are going to learn how to use matrices to create a secret message and de-code a secret message from a classmate.

First they need to learn to compute the inverse of a simple matrix A (provide this matrix to be certain it has an integer inverse.) I prefer a three-by-three, but this can also be done with any size matrix – even a two-by-two. Next, they create their own short message and code it using numbers to represent the letters of the alphabet (A=1, B=2, etc., with 0=space). This coded message should be written into a matrix form, filling in one row at a time (the number of columns MUST match the number of rows in matrix A.) If the secret message does not fill the last row add zeros for spaces. Now, multiply the message matrix by matrix A (with matrix A on the right.)

Message: 7 15 0 21 14 20 0 5 1 7 12 5 19

$\displaystyle \left[ \begin{array}{ccc} 7 & 15 & 0 \\ 21 & 14 & 20 \\ 0 & 5 & 1 \\ 7 & 12 & 5 \\ 19 & 0 & 0 \end{array} \right] \left[ \begin{array}{ccc}3 & 1 & 3 \\ 7 & 10 & -3 \\ 8 & 5 & 5 \end{array} \right]$

This will result in your encoded message:

$\displaystyle \left[ \begin{array}{ccc} 126 & 157 & -24 \\ 321 & 261 & 121 \\ 43 & 55 & -10 \\ 145 & 152 & 10 \\ 57 & 19 & 57 \end{array} \right]$

Now have each student pass this encoded message to another student. Each student must use the inverse of matrix $A$ to de-code the message!

Have them multiply this message matrix by $B A^{-1}$ with the inverse on the right. They will get the de-coded Message matrix. From this they can discover the message!

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Written as an engage:

We are going to begin with a short video today!

(Published on Feb 21, 2013)

This video introduces the Computer Graphics chapter of the “Computer Science Field Guide”, an online interactive “textbook” about computer science, written for high school students. The guide is free, and is available from cosc.canterbury.ac.nz/csfieldguide/ . This video may be downloaded if you need to play it offline.)

What did you notice about the movement of the objects in the video? Does this movement – rotation, position, size – remind you of anything you have done in math class before? What happened to the graph of a function when we multiplied the x value? What about when we multiplied the y value? What happened when we added or subtracted a number to x or y? Do these transformations of functions move in a similar manner as the computer graphics in the video? (Of course, the video shows three-dimensional movement while our graphs only showed two-dimensional movement.)

So what kind of transformations do you think are used to create computer graphics? The graphics you see in your video games, in the movies, on TV, in flight simulators for training pilots, and in many other applications are all created with the transformations of matrices. Matrix multiplication is used in computer graphics to size and scale objects as well as rotate and translate them. Today we are going to learn to compute a special matrix transformation – the inverse of a matrix!

How can technology be used to effectively engage students with this topic?

After the students have learned how to calculate the inverse of a 3 x 3 matrix by hand, you could tell them they are now going to calculate the inverse of a 4 x 4 matrix. After they all roll their eyes and groan, you can ask if they would rather do the calculations by hand or on their graphing calculators.

Now you can introduce a method for entering the data into a calculator (such as the TI83 or TI84.) Since many graphing calculators can handle large matrices, the matrix and the identity matrix can be entered together as a 4 x 8 matrix. By using the “rref(” application, the inverse matrix will automatically be calculated. Another way to calculate the inverse is to enter the matrix then press the x-1 key.

However, you may want to wait before teaching this “short-cut” method. You may choose to have the students enter the 4 x 8 (matrix and identity matrix) and show them how to do the row operations on the calculator. This is useful in helping them see the steps involved in the calculation (and tortures them just a little.)