# Motivating the study of matrices

In Brazil's curriculum students are taught matrices in high school. Here, however, there is no linear algebra or pre-calculus, therefore matrices end up being just tables with lots of "arbitrary" definitions.

Is there a way to motivate the definitions without teaching them things not on the curriculum?

• Do the students know some analytic geometry with vectors beforehand? – quid Nov 6 '14 at 9:24
• @quid no... they have a bit later, but without vectors. they do know vectors from physics, but more as "arrows". – Lucas Virgili Nov 6 '14 at 17:17
• Not sure if this will be interesting for your students, but if they are into computers or video games, all computer graphics boil down to matrices. You don't have to get into the nitty gritty but just talking about the connection could help to engage some of them. You can do "toy" examples using rotation, translation, and reflection matrices just like they are used in every video game since vector graphics. I discussed this with my students and they really seemed to think it was a cool application of matrices and helped get them interested – celeriko Nov 12 '14 at 17:22
• If your students can program, then let them play around with pixel matrices. Check out this: jeremykun.com/2012/01/01/random-psychedelic-art – Aivar Dec 7 '14 at 11:34
• Also see codingthematrix.com – Aivar Dec 7 '14 at 11:35

Even without explicitly introducing the language of "linear maps", "vectors", and so on, you can still develop matrices as a shorthand for such maps, thought of as exchange rates.

Example:

Machine A can make 3 sprogs and 2 sprakets a day. Machine B can make 1 sprog and 3 sprakets a day. We summarize this data in a table of values:

$$\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix}$$

Where the first column represents machine A and the second column machine B, the first row sprogs, and the second row sprakets.

Now If my company buys $7$ machine As and $3$ machine Bs, what is my production capacity? This is a reasonable question for a 3rd grader, probably. All the matrix does is provide a structure for solving such a problem systematically, as:

$$\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 7 \\ 3 \end{bmatrix}$$

Multiplying a matrix by a vector should be defined to make the interpretation above valid.

Now say each sprog sells for $5$ and each spraket sells for $4$ dollars. Then we can get a new table telling us how much money machine A makes and how much money machine B makes. Again, this is a "third grade" problem, and we just introduce new notation for the problem:

$$\begin{bmatrix} 5 & 4\end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix}$$

Another example might be that each sprog lets another company build 2 cars and 3 buses, while a spraket lets them build 4 cars and 2 buses. So ultimately, to figure out how machine A and B relate to cars and buses, you can perform:

$$\begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix}$$

At each stage you should be carefully tracking the meaning of these objects. The columns each represent one of a set of inputs, and each row represents an output. The entry at the $i^{th}$ column and $j^{th}$ row is the quantity of output $j$ produced by $1$ input $i$. Matrix multiplication corresponds to figuring out a new table if you are given two tables where the outputs of one are the inputs of the other.

That is to say, I suggest teaching the linear algebra in a particularly down to earth way, which is unlikely to intimidate students (or other teachers!) with words like "linear transformation" or "vector space".

• I am afraid I do not know how to motivate determinants at this level, however. I know that, in the US at least, I was forced to learn determinants of $2\times 2$ and $3 \times 3$ matrices with no application or motivation in about the 10th grade. – Steven Gubkin Nov 6 '14 at 22:26
• I like motivating matrices as a way to organize sums and products. Can you think of any objects to substitute for "sprig" and "spraket" which make the numbers in the matrix plausible? Or is there a better example than machine production? – user173 Nov 7 '14 at 3:32
• @MattF. Any constant exchange rate with more than one input and output will work. Pill A could deliver $3 mg$ aspirin and $2 mg$ acetaminophen, while Pill B could deliver $1 mg$ aspirin and $3 mg$ acetominophen for example (there are many such combination drugs on the market). You want to determine how much of each pill to give to achieve $10 mg$ aspirin and $20 mg$ acetaminophen. That amounts to solving $\begin{bmatrix} 3 &1\\2& 3\end{bmatrix}\begin{bmatrix} x \\y\end{bmatrix} = \begin{bmatrix} 10 \\ 20\end{bmatrix}$. – Steven Gubkin Nov 7 '14 at 3:41
• In my high-school senior year in Israel, determinants were introduced as a tool to be used for calculating areas and volumes in analytic geometry. It wasn't really a part of curriculum, but more a kind of hack/shortcut and I found them to be really useful. – yoniLavi Nov 7 '14 at 20:29
• @Steven Gubkin: symbols of units, such as milligram (mg), are written in Roman type. For an educated person, “$mg$” is weight. – Incnis Mrsi Aug 24 '15 at 11:28

Maybe you can try solving sets of linear equations (2 × 2 or 3 × 3) as a motivation example for matrices and their basic manipulations.

• In HS, using reduced row echelon form/ augmented matrices to solve systems of equations was definitely my biggest motivator for learning matrices – celeriko Nov 17 '14 at 16:06

Short of changing the curriculum :-), maybe you can use magic squares, or even Sudoku, to help motivate. See, e.g., "From Magic Squares to Sudoku". Here is Dürer's $4 \times 4$ magic square: • But this is not what matrices do. When I saw matrices like this in school I said to myself... what is the point of this?? – JP McCarthy Nov 7 '14 at 13:39
• @JpMcCarthy: Well, at least you could introduce the trace of a matrix. There is also an interesting connection to the rank and to the eigenvalues of a magic square. But I am stretching matters; your point is well-taken. – Joseph O'Rourke Nov 9 '14 at 21:31

One of the favorite mathematical activities of many students is "cancellation." An example is when you are asked to reduce a complicated fraction, and you can do so by "factoring out" a large number from both numerator and denominator, leaving "lowest terms."

If you can convince students that matrices are really representative of systems of equations, they can have a lot of fun reducing the systems of equations to lowest terms. Using "low echelon row reduction" for instance, you can create a diagonal matrix, where one equation is a number of x, the equation above it relates to only x's and y's, etc.

Another thing to do is to show how matrices determine polyhedrals in space. And how the related determinants "reduce" to the volumes of these polyhedrals.

Surely you teach complex numbers in the curriculum.

One thing to do with matrices is to use them to create other mathematical objects, or at least a model of the object. In particular, we can use $$2 \times 2$$ real matrices of the form $$\left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]$$ to model the complex number $$x+iy$$. Matrix addition and multiplication then correspond naturally to the addition and multiplication of complex numbers and the complex number $$1$$ corresponds naturally to the matrix $$\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]$$ etc. In addition, in this discussion you can either look at the formula $$\frac{1}{z} = \frac{\bar{z}}{z\bar{z}}$$ as a method to find the multiplicative inverse of a complex number, or you can use the $$2\times 2$$ inverse formula (if that is in the scope of the discussion).

Try to handle and solve for the $$x$$'s conceptually this system, or this one

$$A\mathbf x = \mathbf b$$

where $$A$$ is a $$K\times K$$ matrix, and $$\mathbf x, \mathbf b$$ are $$K \times 1$$ vectors, $$K=10^6$$. Hmm, why not set $$K$$ equal to one trillion? Or a hundred thousand trillion.

And when the Matrix Algebra interrelations start to become more and more intricate and impressive, beyond the basic level of solving a linear system of equations, and still the dimensions can be as large as we want, this is how I fell for Linear Algebra and matrices: once I realized that this dance remains elegant irrespective of the size of the entities we dance with (while also being able to visualize somehow the enormity of them -in most other mathematics, the object dimensions are a bit too abstract for most people).

It's like Circus Human Acrobatics perfectly performed by elephants. Or galaxies.