Engaging students: Computing the determinant 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 Brendan Gunnoe. His topic: computing the determinant of a matrix.

How can this topic be used in your students’ future courses in mathematics or science?

When students learn about the determinant of a matrix, they only learn about computing it and don’t learn about the applications of the determinant or what they signify. One interesting use of the determinant is finding the eigenvectors of a matrix. A visual understanding of what an eigenvector is can be done by showing what a matrix does to the any generic vector, and what it does to the eigenvectors. For a generic vector that is different from an eigenvector, the matrix knocks the vector off the span of the original vector. What makes an eigenvector special is the fact that the matrix transformation keeps the eigenvector on its span but rescales said eigenvector by its eigenvalue. For example, take the matrix

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right]$ .

This matrix’s eigenvectors are $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$ with eigenvalues 8 and 2 respectively. That is,

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right] \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] = \left[ \begin{array}{c} 8 \\ 8 \end{array} \right] = 8 \left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$

and

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right] \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] = \left[ \begin{array}{c} 2 \\ -2 \end{array} \right] = 2 \left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$ .

Eigenvectors have many useful applications in future math and science classes including electronics, linear algebra, differential equations and mechanical engineering.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The YouTube channel 3Blue1Brown has a fantastic video on determinates and linear transformations. Grant, the channel owner, uses animations to visualize what a matrix transformation does to the plane . He starts by showing what a transformation does to a single square then shows why the change of change of that one area shows what happens to the area of any region. He also gives multiple explanations for what a negative determinate means in terms of orientation of the axes. Then he explains what happens when the determinate is 0. All of this is already extremely useful for understanding what a 2×2 matrix does, but Grant continues and extends all the same things for 3×3 transformations. Lastly, Grant gives a few explanations on why the formula for the determinate is what it is and poses a small puzzle for the viewer to contemplate. This video explains what and why we use determinates and how they can be useful all while showing pleasing visual examples and other explanations.

What interesting word problems using this topic can your students do now?

Using determinates to see if a set of vectors is a basis.

The determinant can tell you when a matrix squishes space into a lower dimensional space like a line or a plane. Thus, taking the determinate of a matrix composed of a set of vectors tells you if those vectors are a basis for the matrix’s dimension.

Part 1. A 3D printer’s axes are set up in such a way that the print head can only travel in the direction $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} -1 \\ 1 \end{array} \right]$ . Assume that the place where the print head is right now is the origin $\left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$ . Can you move the print head to the location $\left[ \begin{array}{c} x \\ y \end{array} \right]$ and $\left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$ by only moving in the directions of $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} -1 \\ 1 \end{array} \right]$ ?

Hint: Try to solve $\left[ \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \end{array} \right] = \left[ \begin{array}{c} x \\ y \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \end{array} \right]$ ?

Part 2. Next time you turn on your 3D printer, one of the motor’s broke and now the print head can only move in the direction of $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$ . Assume that the place where the print head is right now is the origin $\left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$ . Can you move the print head to the location by only moving in the direction of $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$ ?

Hint: Try to solve $\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right] \left[ \begin{array}{c} a \\ b \end{array} \right] = \left[ \begin{array}{c} x \\ y \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \end{array} \right]$ ?

Part 3. You buy a new 3D printer that it can move in all three directions: up/down, left/right, forward/backwards. When you test out the movement of the print head, you see that each axis moves in the directions of $\left[ \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right]$ , $\left[ \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right]$ , and $\left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right]$ . Can you use your new 3D printer to go to any location $\left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ , inside the printing space?

Hint: Think about solving $\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \\ c \end{array} \right]$ ? How do you know?

Part 4. Your little sibling messed around with your new 3D printer and now it moves in the directions $\left[ \begin{array}{c} 1 \\ 0 \\ 1 \end{array} \right]$ , $\left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right]$ , and $\left[ \begin{array}{c} 2 \\ 1 \\ 1 \end{array} \right]$ . Is your 3D printer able to get to some point $\left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ inside the printing space as is, or do you need to fix your printer?

Hint: Think about solving $\left[ \begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \\ c \end{array} \right]$ ? How do you know?

Engaging students: Solving linear systems of equations with matrices

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

This student submission comes from my former student Andrew Sansom. His topic, from Algebra II: solving linear systems of equations with matrices.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

The Square in Downtown Denton is a popular place to visit and hang out. A new business owner needs to decide which road he should put an advertisement so that the most people will see it as they drive by. He does not have enough resources to traffic every block and street, but he knows that he can use algebra to solve for the ones he missed. In the above map, he put a blue box that contains the number of people that walked on each street during one hour. Use a system of linear equations to determine how much traffic is on every street/block on this map.

HINT: Remember that in every intersection, the same number of people have to walk in and walk out each hour, so write an equation for each intersection that has the sum of people walking in is equal to the number of people walking out.
HINT: Remember that the same people enter and exit the entire map every hour. Write an equation that has the sum of each street going into the map equal to the sum of each street going out of the map.

Solution:

1. Build each equation, as suggested by the hints.

2. Rewrite the system of simultaneous linear equations in standard form.

3. Rewrite the system as an augmented matrix

4. Reduce the system to Reduced Row Echelon Form (using a calculator)

5. Use this reduced matrix to find solutions for each variable

This gives us a completed map:

Clearly, the business owner should advertise on Hickory Street between Elm and Locust St (Possibly in front of Beth Marie’s).

B1. How can this topic be used in your students’ future courses in mathematics or science?

Systems of Simultaneous Linear Equations appear frequently in most problems that involve modelling more than one thing at a time. In high school, the ability to use matrices to solve such systems (especially large ones) simply many problems that would appear in AP or IB Physics exams. Circuit Analysis (including Kirchhof’s and Ohm’s laws) frequently amounts to setting up large systems of simultaneous equations similar to the above network traffic problem. Similarly, there are kinematics problems where multiple forces/torques acting on an object that naturally lend themselves to large systems of equations.

In chemistry, mixture problems can be solved using systems of equations. If more than substance is being mixed, then the system can become too large to efficiently solve except by Gaussian Elimination and matrix operations. (DeFreese, n.d.)

At the university level, learning to solve systems using matrices prepares the student for Linear Algebra, which is useful in almost every math class taken thereafter.

D4. What are the contributions of various cultures to this topic?

Simultaneous linear equations were featured in Ancient China in a text called Jiuzhang Suanshu or Nine Chapters of the Mathematical Art to solve problems involving weights and quantities of grains. The method prescribed involves listing the coefficients of terms in an array is exceptionally similar to Gaussian Elimination.

Later, in early modern Europe, the methods of elimination were known, but not taught in textbooks until Newton published such an English text in 1720, though he did not use matrices in that text. Gauss provided an even more systematic approach to solving simultaneous linear equations involving least squares by 1794, which was used in 1801 to find Ceres when it was sighted and then lost. During Gauss’s lifetime and in the century that followed, Gauss’s method of elimination because a standard way of solving large systems for human computers. Furthermore, by adopting brackets, “Gauss relieved computers of the tedium of having to rewrite equations, and in so doing, he enabled them to consider how to best organize their work.” (Grcar J. F., 2011).

The use of matrices in elimination appeared in 1895 with Wilhelm Jordan and 1888 by B.I. Clasen. Since then, the method we use today has become commonly attributed to Jordan and commemorated with the name “Gauss-Jordan Method”.
References:
DeFreese, C. (n.d.). Mixture Problems. Retrieved from University of Missouri-St. Louis–Department of Mathematics and Computer Science: http://www.umsl.edu/~defreeseca/intalg/ch8extra/mixture.htm
Grcar, J. F. (2011, May). Historia Mathematica–How ordinary elimination became Gaussian elimination. Retrieved from ScienceDirect: https://www.sciencedirect.com/science/article/pii/S0315086010000376
Grcar, J. F. (n.d.). Mathematics of Gaussian Elimination. Retrieved from American Mathematical Society: https://www.ams.org/notices/201106/rtx110600782p.pdf

Engaging students: Adding, subtracting, and multiplying matrices

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 Diana Calderon. Her topic, from Algebra: adding, subtracting, and multiplying matrices.

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

As a teacher I would do a foldable activity in which the students will have to fill in the blank in the front of the foldable that would allow them to discover how addition, multiplication and subtraction work for matrices. Once they open the foldable, they would have to do different examples and get to also create one. Out of the problems that the students create, as a teacher, I would choose one of each and allow them to go up to the board and explain how they did it and address any misconceptions that may have happened when they were discovering how the concepts work. I plan on doing my foldable with color coding so that the students can see where the numbers in the columns and rows changed when the matrices were added, multiplied, or subtracted, I will most likely limit the matrices to vary from 2×1,2×2,2×3,3×2,and 3×3.

How does this topic extend what your students should have learned in previous courses?

– The topic of adding, subtracting and multiplying matrices allows students to extend their knowledge when it comes to adding, multiplying or subtracting polynomials. I can show the students how a polynomial is similar to a 1×1 matrix. Another subject that they may see something similar to matrices would be in Biology with the punnett squares, it can be as basic as doing it for one generation or two and then go from there on. As said in the article “Use of Matrices to Determine Genetic Probability” by Andrew Almendarez, “Through prescribed manipulations and interpretations matrices can be used to represent and solve physical problems. One such problem is finding the probability of a certain genotype within a population over multiple generations.”, this also ties into probability which they most likely learned the previous year. It would be good to tell them that if they are interested in the medical lab field for example, “trying to breed cows that produce the most milk. If cows of a certain genotype were known to produce more milk than others it is useful to know how many cows of that genotype there will be after a number of generations, and what will maximize the proportion of that genotype in the future. This is where the Punnett is used in conjunction with matrices”.

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

The topic of matrix multiplication came up when I looked in the news. Recently, apple has been one of the most popular brands when it comes to computers, cellular devices, TV, ear phones, etc. With that being said, every year or so they release a new “it” item. This month they are releasing the new iPhone 11, which overall in my opinion is the best cellular device one can get, it has a sleek professional design, great camera, a huge amount of storage embedded within itself and many other useful resources that one utilizes in their everyday life. In the article “iPhone 11: Apple’s A13 Bionic Chip Enables More Than Just Faster Speed” it mentions matrices and how “The A13 Bionic has a whopping 8.5 billion second-generation seven-nanometer transistors, up from 6.9 billion in the previous generation. It can perform one trillion operations per second, thanks in part to new machine learning accelerators that can run matrix multiplication six times faster.”. For me it is amazing to know just how fast these devices can calculate anything and everything one wants to find out instead of doing it by hand.

Citations:
• Use of Matrices to Determine Genetic Probability
https://www.academia.edu/20442574/Use_of_Matrices_to_Determine_Genetic_Probability

• iPhone 11: Apple’s A13 Bionic Chip Enables More Than Just Faster Speed
https://www.inverse.com/article/59239-a13-chip-faster-more-efficient

Matrix Jokes

A lot more Matrix jokes can be found at https://mathwithbaddrawings.com/2018/03/07/matrix-jokes/

Engaging students: Solving linear systems of equations with matrices

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

This student submission comes from my former student Danielle Pope. Her topic, from Algebra II: solving linear systems of equations with matrices.

B2. How does this topic extend what your students should have learned in previous courses?

Based off of the TEKS, matrices are introduced in Algebra 2. In previous math courses, students are already going to learn basic arithmetic from elementary school and solving equations in middle and high school. By the time students get to high school, they should have solving single equations down. This concept is then expanded with a system of equations, which is taught with the help of matrices. A matrix is just an “array of numbers” so that’s why this method of solving can be used with linear equations. Once the matrix is set up there are 2 main ways to solve for the solutions. The one I will be discussing is reduced row echelon form. This method of solving systems utilizes the basic arithmetic that students already know. There are 3 row operations that students already know how to use in general not related to matrices. Those are multiplying a row by a constant, switching two rows, and adding a constant times a row to another row. Even though these specific operations are used for matrices, kids have seen how to multiply 2 constants or variables, switching variables, and adding constants or variables in their previous courses. Matrices just add another element to their basic arithmetic abilities.

D4. What are the contributions of various cultures to this topic?

Matrices have been around for much longer than some people may realize. One of the earliest civilizations that matrices were traced back to were the Babylonians. This was just one of the many contributions that they contributed to mathematics. The Chinese wrote a book, Nine Chapters of the Mathematical Art, Written during the Han Dynasty in China gave the first known example of matrix methods”. During the same era, around 200 BC, a Chinese mathematician Liu Hui solved linear equations using matrices. In the 1800s, Germany started taking a look at matrices. German mathematician, Carl Jacobi, brought the idea of determinants and matrices into the light. Carl Gauss, another German mathematician, took this idea of determinants and developed it. It wasn’t until Augustin Cauchy, a French mathematician, used and defined the word determinant how was use it today. James Sylvester, an English mathematician, “used the term matrix in 1850”. Sylvester also worked with mathematician Arthur Cayley who “first published an abstract definition of matrix” in his memoir on the Theory of Matrices in 1858. This final definition of a determinant is still used today in classrooms to help solve complex system of equations.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

In a classroom today, students should be able to access use of a graphing calculator. The matrix feature on these can easily check the work of students just learning how to row-reduce or solve for determinants and inverse matrices. In the classroom, I would use this technology like a race for the right answer to get them engaged in matrices. Give students an easy 2-equation system and have them solve for the variables. Each new problem add an equation or add a variable. While students are solving by hand, the teacher will be using the calculator to see which person can get the answer first. Overtime the problems will be too daunting to do by hand so students will be more engaged to learn this faster shortcut using the calculator. Another resource that can be used out of the classroom is Khan Academies’ videos on solving system of equations with matrices. These videos can be used to fill in any gaps if students have questions at home. These videos can also be used as the lecture in a flipped classroom environment.

References

https://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html
http://www.sparknotes.com/math/algebra2/matrices/section4.rhtml
http://math.nie.edu.sg/bwjyeo/it/MathsOnline_AM/livemath/the/IT3AMMatricesHistory.html

http://math.nie.edu.sg/bwjyeo/it/MathsOnline_AM/livemath/the/IT3AMMatricesHistory.html
http://www.storyofmathematics.com/mathematicians.html

https://www.khanacademy.org/math/precalculus/precalc-matrices/solving-equations-with-inverse-matrices/v/matrix-equations-systems

Nilpotent Matrix

Source: https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549.1073741828.663847933632036/1603583536325133/?type=3&theater

Engaging students: Adding, subtracting, and multiplying matrices

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 Perla Perez. Her topic, from Algebra: adding, subtracting, and multiplying matrices.

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

“Cryptography is the study of encoding and decoding messages. Cryptography was first developed to send secret messages in written form.” Cryptography also uses matrices to code and decode these messages by multiplication and the inverse of them. This, however, can be done by using any operations. By using the worksheet below as a foundation for an activity, teachers can have students act like hackers to engage students in computing different operations with matrices. In this activity, prepare the classroom by dividing it into four sections each with one of the phrases separated on the worksheet. Display the message (numerically) that is to be coded. Display the alphabet with corresponding number somewhere visible for students to have references throughout the activity. The instructions given are:

Students are to get into four groups (more groups can be added for larger classrooms by making the phrase longer).
Students are given an index card with the matrix [2, 7; 13, 5]
Students are to add the matrix on each station to the the matrix on the card.
When completed students must go change the message on the broad with the code.

When the students finish coding the message they can continue developing their skills by having them do this in the beginning of class throughout the lesson plan period. As the lesson progresses the teacher can change the phrase and require different operations to be made to either code or decode or even come up with their own message. With this activity the teacher gets the opportunity to see how the students choose to add the matrices together.

Click to access using_matrices_in_cryptography_intro.pdf

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

In today’s society we have access to a plethora of technology that can aid us in our everyday lives. There are so many ways one can learn something with different methods and from different people. The best part about the technology that we have access to is we can be manipulative to fit the needs of our students. When students get to the topic of adding, subtracting and soon multiplying matrices, they should be familiar with what a matrix is, the dimensions of one, and how to solve linear system with them. At this point it is a good a time to bring in a game into play like this one:

http://www.intmath.com/matrices-determinants/matrix-addition-multiplication-applet.php.

In this game the player chooses an operation such as adding, subtracting, multiplying by another matrix or scaler, and its dimensions. When a certain operation is chosen such as multiplication, it only allows the player to choose any size matrix but then spits out one with specific number of rows to multiply it with. The teacher can play this game with their students in any way they sit. The purpose is to get students thinking why and how the operations are working. From there the teacher can introduce the new topic.

Resources:

http://www.intmath.com/matrices-determinants/matrix-addition-multiplication-applet.php

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

So many times students don’t understand that what they learn in class is used in everyday life, but teachers can give students the resources and knowledge to see applications of their work. In the video below, it shows different ways matrices can be applied. For instances the operations of matrices are used in a wide variety of way in our culture.

The main one can be in computer programming and computer coding, but they are also seen in another places such as dance and architecture. “In contra dancing, the dancers form groups of four (two couples), and these groups of four line up to produce a long, two-person-wide column” and where each square that is created is a formed by two pairs. Like the video had said, matrices can be used to analyze contra dancing. This can be done by having squares and multiplying them creating different types of configurations. By creating different groups and formations, essentially it is using different operations to create different matrices to.

Resources:

References:

“Common Topics Covered in Standard Algebra II Textbooks.” Space Math @ NASA. NASA, n.d. Web. 18 Sept. 2015.

Knill, Oliver. “When Was Matrix Multiplication Invented?” When Was Matrix Multiplication Invented? Harvard, 24 July 2014. Web. 18 Sept. 2015.

Smoller, Laura. “The History of Matrices.” The History of Matrices. University of Arkansas at Little Rock, Apr. 2001. Web. 18 Sept. 2015.

Engaging students: Adding, subtracting, and multiplying matrices

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 Joe Wood. His topic, from Algebra: adding, subtracting, and multiplying matrices.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

One interesting real world problem for matrix operations can be found in Chapter 4.1.3 at http://spacemath.gsfc.nasa.gov/algebra2.html. The problem deals with astronomical photography. It starts by explaining the process by which NASA gets its images and relates the process of taking the pictures from blurry to clear using matrices. The problem goes as follows:

For a way to engage students who are not interested in astronomy, and to allow students to learn more on their own time of the uses, a homework assignment could be for them to find places other than NASA that this process could be used.

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

“Nine Chapters of the Mathematical Art”, an ancient book that dates between 300 BC and AD 200, gives the first documented use of matrices. Even though matrices were used as early as 300 BC, the term “matrix” was not used until 1850 by James Joseph Sylvester. The term matrix actually comes from a Latin word meaning “womb”.

Below is a list published on the Harvard website of important matrix concepts and the years they were introduced.

200 BC: Han dynasty, coefficients are written on a counting board [6]

1545 Cardan: Cramer rule for 2×2 matrices. [6]

1683 Seki and Leibnitz independently first appearance of Determinants [6]

1750 Cramer (1704-1752) rule for solving systems of linear equations using determinants [8]

1764 Bezout rule to determine determinants

1772 Laplace expansion of determinants

1801 Gauss first introduces determinants [6]

1812 Cauchy multiplication formula of determinant. Independent of Binet

1812 Binet (1796-1856) discovered the rule det(AB) = det(A) det(B) [1]

1826 Cauchy Uses term “tableau” for a matrix [6]

1844 Grassman, geometry in n dimensions [14], (50 years ahead of its epoch [14 p. 204-205]

1850 Sylvester first use of term “matrix” (matrice=pregnant animal in old french or matrix=womb in latin as it generates determinants)

1858 Cayley matrix algebra [7] but still in 3 dimensions [14]

1888 Giuseppe Peano (1858-1932) axioms of abstract vector space [12]

B1. How can this topic be used in your students’ future courses in mathematics or science?

Matrices and matrix operations are used in many math classes from Algebra and Calculus, to Linear Algebra and beyond. So any student interested in studying any discipline of Engineering or mathematics should become very familiar with matrices since they are used in a wide variety of ways (one way is seen above). Matrices are also useful in other courses as well. In Chemistry, matrices can be used for balancing chemical equations. In Physics, matrices can be used to decompose forces. Even in ecology or biology classes, matrices can be crucial. A great example would be studying animal populations under given conditions.
One hope in giving so many brief examples is that a student who cares nothing about the topic of matrices would here about a topic they are interested in (say animals) and that would spark questions into how or why matrices are useful. And of course, when dealing with matrices, addition subtraction, and multiplication of matrices follows closely behind.

References:

“Common Topics Covered in Standard Algebra II Textbooks.” Space Math @ NASA. NASA, n.d. Web. 18 Sept. 2015.

Knill, Oliver. “When Was Matrix Multiplication Invented?” When Was Matrix Multiplication Invented? Harvard, 24 July 2014. Web. 18 Sept. 2015.

Smoller, Laura. “The History of Matrices.” The History of Matrices. University of Arkansas at Little Rock, Apr. 2001. Web. 18 Sept. 2015.

Deciphering recommendation engines

From the video’s description: “Data scientist Cathy O’Neil provides a glimpse of the methods that Netflix, Google, and others apply to recommend or offer to users selections based on their apparent interests.” This is a non-intuitive but real application of linear algebra.

Engaging students: Finding the inverse of a matrix

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.)