# 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?

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