Is there a good way to explain determinants in an elementary linear algebra class?

Question

Many colleges offer an an elementary linear algebra class for sophomore math, science, and economics majors. Such a class typically covers a chapter on determinants, including the following aspects:

• Cofactor expansion using any row or column.

• How the determinant changes when you perform row operations.

• The fact that a determinant can be viewed as the volume of a parallelepiped.

• The fact that $\det(AB) = \det(A) \det(B)$.

• Cramer's rule.

• The formula for the inverse of a matrix using the classical adjoint.

Usually this material needs to be covered relatively quickly, e.g. in a week or so of class.

My question is:

Is there any way of covering determinants quickly that fully explains all of this material?

In my experience, it's usually possible to give explanations for some of this material, but there's no one way of looking at determinants that provides a simple explanation for all of it.

Answer 1

I subscribe to the down with determinants school of thought. In an elementary linear algebra course, particularly for non-mathematicians, then I think that determinants are as useful as your appendix[1]. Nevertheless, they are on the syllabus so I have to teach them.

Here's what I do. I tell the students the truth: that determinants used to be a key part of doing computations with matrices but now there are better methods, however that they will still need to compute some determinants on an exam. Then I tell them how to compute them using Gaussian Elimination, and the tricks for 2x2 and 3x3 (emphasising that they are tricks, and that there is no such trick for 4x4 or higher). If time, I might mention the relationship to volume but I wouldn't regard it as important particularly as up to this point we won't have talked about geometry at all. I certainly wouldn't bother with Cramner's rule.

In my current course, I have four hours set aside for determinants. The above takes a little over one. I use the rest of the time for more useful things like going back over topics that the students have had issues with earlier in the semester.

Here are a couple of useful nuggets to reinforce the down with determinants line.

1. Myth: "Determinants are useful to figure out if a matrix is invertible or not."

   Well, in theory, maybe. But in practice, no. How do we compute the determinant? Generally, by using Gaussian Elimination but remembering row swaps and row scales. But Gaussian Elimination tells us directly whether or not the matrix is invertible.

2. Myth: "Determinants are useful for finding eigenvalues."

   Again, this is true in theory, but not in practice. It's fine with 2x2, and with 3x3 if you give them a running start (such as an obvious root). But any higher and you're in dubious territory.

   Indeed, a few years ago I looked up how one of the major mathematical pieces of software found roots of polynomials. What I found surprised me, and put the final nail in determinant's coffin. To find the roots of an arbitrary polynomial, this program figures out a matrix with that as its characteristic polynomial. Then does an iterative method to find the eigenvalues of that matrix, and returns them as the roots of the polynomial. I leave it as an exercise to the reader to see how this shows the idiocy of teaching determinants as a method for finding eigenvalues.

[1] I want to make it clear that this is about determinants in such a course. As a differential topologist, I'm fully aware of the usefulness of determinants in other arenas. But the time to introduce them is in those other arenas, not in an elementary linear algebra course.

Answer 2

I spend 4 (50 minute) days on determinants and one homework assignment. This is a course mostly made up of engineers and some economists/social scientists, taught out of Bretscher's book. Here is what I said the most recent time I did it:

Day One Derive how $2 \times 2$ determinants come up in computing areas of parallelograms and triangles, in inverting $2 \times 2$ matrices and in checking whether $2 \times 2$ matrices are invertible. Assert without proof that the $3 \times 3$ determinant formula has similar properties. State the multilinear properties of determinant and work through them from as many perspectives as I can: How can we prove them from the formulas? Why is area multilinear (up to sign)? The area discussion again with a topologically different configuration of vectors, so they can see the signs popping up. If $(u,v,w)$ and $(u,v,w')$ are dependent, why should $(u,v,a w+ a' w')$ be dependent for scalars $a$ and $a'$?

Day Two Assert that there is a multlinear formula, like determinant, for $n \times n$ matrices. Show how to compute it using row reduction just from multilinearity, and how to take short cuts for upper triangular or block upper triangular matrices.

Day Three Give the formula in terms of a sum of $n!$ permutations but emphasize that we almost never want to use it for $n>3$. Show that it is multlinear and alternating; don't worry if they don't really follow this. Emphasized that I am only presenting this so that they know determinant exists; they shouldn't think in terms of this formula.

Day Four Determinants as volume. To some extent, this repeats day 1, but by now they are used to the formulas involved. Talk about why you might want to divide a region into a nonsquare grid and sum to numerically compute an integral, so they need to know volumes of the grid pieces. (Multivariable calc isn't formally a prereq, so I can't do anything deep with multivariate integrals, but a lot of them have it.) Describe the geometry of the formula $\det (AB) = (\det A)(\det B)$. (I also prepared a handout with a rigorous proof for those who were interested.)

Of your list, I never do Cramer's rule or adjugates. They are useful for algebraists (like me!) but their applied use is made obsolete by computers. I spend a lot of time on using multilinearity and on geometric thinking. Cofactor expansion isn't presented as a separate subject, but when I talk about multilinearity, I do examples like: $$\det \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} = \det \begin{pmatrix} 1 & 0 & 0 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} +\det \begin{pmatrix} 0 & 2 & 0 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} + \det \begin{pmatrix} 0 & 0 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} =$$ $$\det \begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} - 2 \det \begin{pmatrix} 4 & 6 \\ 7 & 9 \end{pmatrix} +3\det \begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix} . $$

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Update: Thanks to COVID, I now have lecture videos to go with these! I've rearranged the order since I wrote this answer.

Video 1 (slides): The properties of the determinant. More or less day 2 above.

Video 2 (slides): Geometry of determinants. More or less days 1 and 4.

Video 3 (slides): The formula for the determinant. More or less day 3.

This class had prerecorded lecture videos (like these), plus interactive Zoom sessions with group work and discussion. If I recall correctly, a number of the motivational pictures in video 2 were drawn informally on zoom on the first day.

Answer 3

I'd say it's just plain tough to get through all this quickly. For an elementary class I typically dump out most proofs of why various properties hold and focus more on how the ideas hang together.

I advocate starting with $1 \times 1$, $2 \times 2$, and $3 \times 3$ (just expanding along the top row for the last one). Then show off the $3 \times 3$ determinant trick (down diagonals minus up diagonals).

Then discuss how $2 \times 2$ determinants give signed areas of parallelograms (the sign encoding orientation: clockwise vs. counter-clockwise). $3 \times 3$ giving signed volumes of parallelepipeds (the sign encoding orientation: right-hand vs. left-hand rules). Then mention $n \times n$ give $\pm$ $n$-dimensional volume of the corresponding parallelotope.

Next, general row/col expansions (picking out examples with lots of zeros to make it worth while). Briefly mention how one can use permutations to define the determinant. Maybe show off how this works for the $2 \times 2$ case. In a class where proofs are appropriate, maybe indicate how decomposing the symmetric group allows one to get these various expansions.

Next, without doing precise proofs, I mention multi-linearity and skew-symmetry/alternating properties. Then show how elementary operations effect the determinant (flowing from the properties). Once this is done I like to sketch out how one can compute determinants by doing a forward pass of Gauss-Jordan elimination (do type I & III operations - no scales - until upper-triangular, multiply diagonal, adjust the sign according to the number of swaps). Mention how much faster this is when we have a sizable matrix.

Elementary operations yield an easy proof that the determinant is multiplicative. Make sure to mention why $\det(cA) = c^n\det(A)$ not $c\det(A)$.

I then like to quickly sketch Cramer's rule and how the classical adjoint inverse formula works. Emphasizing how the only division performed when finding an inverse is division by the determinant. Mention how an integer matrix with determinant $X \not=0$ has an inverse involving fractions with denominator $1/X$. [Nice professors give matrices whose determinants are $\pm 1,\pm 2$. :)]

Mostly as I go through this I skip any detailed proof that involves matrix components. If I feel such a proof is warranted, I'll just do the proof for $2\times 2$ or $3 \times 3$ matrices. I think it best serves the students to emphasize the connections between various properties and not the nuts-and-bolts nasty symbol shuffling proofs to get those properties - at least for an elementary class.

Answer 4

1) the definition of the determinant as a volume function easily gives rise to

2) the formula for the determinant using permutation, which easily gives rise to

3) cofactor expansion.

2 and/or 3 give rise to

4) how the determinant changes with elementary row operations, which gives rise to

5) a proof that the determinant is multiplicative.

6) both Cramer's rule and the classical adjoint formula are easily derived directly by computing.

If you have two lecture hours a week then covering all of this may be a little tight, but doable. Certainly if you delegate some of this to an exercise (e.g., Cramer's rule and the adjoint formula, or the proof of the multiplicative property (with some guidance)).

Answer 5

Warning: What follows are the words of someone who has not taught a linear algebra course. Take everything with the appropriate level of salt.

A great theme to try and convey throughout a linear algebra course is that sometimes it is fruitful to try to understand a mathematical object by characterizing properties.

The first example students would probably see in a first linear algebra course is the formula for matrix multiplication. If you give the definition as the formula, it is incomprehensible. Why would anyone ever perform this horrible operation? On the other hand, if you define matrix multiplication as the matrix which results from composing the linear maps they represent, it is totally motivated. This definition seems less "computational", but it is actually just as good for computation.

The determinant provides another great example of something which is best understood through properties which characterize it:

Start with the motivation that for vectors in $\mathbb{R}^n$, we want to define the signed volume of the parallelepiped spanned by an ordered list of vectors $(v_1,v_2,...,v_n)$.

We should have

1. $D(e_1,e_2,...,e_n) = 1$
2. Switching two inputs should negate the output (The determinant is alternating)
3. The determinant is linear in each vector input separately (This requires some carefully drawn pictures)

Now have them break up into groups and try to compute some determinants using these properties. Since the only value they know is on the ordered list of basis vectors, they must proceed by using multilinearity reduce to basis vectors, and then the alternating property to put them into the correct order. For example, you might have them compute

$ \begin{align*} D\left(\begin{bmatrix}2\\3\end{bmatrix},\begin{bmatrix}4\\5\end{bmatrix}\right) &=D\left(2\begin{bmatrix}1\\0\end{bmatrix}+3\begin{bmatrix}0\\1\end{bmatrix},\begin{bmatrix}4\\5\end{bmatrix}\right)\\ &=2D\left(\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}4\\5\end{bmatrix}\right)+3D\left(\begin{bmatrix}0\\1\end{bmatrix},\begin{bmatrix}4\\5\end{bmatrix}\right)\\ &=(2)(4)D\left(\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}1\\0\end{bmatrix}\right)+(2)(5)D\left(\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix}\right)+(3)(4)D\left(\begin{bmatrix}0\\1\end{bmatrix}, \begin{bmatrix}1\\0\end{bmatrix}\right)+ (3)(5)D\left(\begin{bmatrix}0\\1\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix}\right)\\ &=2(5) -3(4)\\ &=-2 \end{align*} $

I really think that this kind of computation, where each step is dictated by the properties of the determinant, is essential. For some reason, I have not really seen this kind of computation in any linear algebra book. It is usually only shown in the general case, as part of the proof that the determinant is characterized by these properties, and I think students often "turn off" for proofs.

This is as far as I wanted to go in my comment in the other thread that the "computation of the determinant is clear from its characterizing properties". I do think covering all of the things you mention in a week is probably a bit much. So far, the above presentation has covered the volume aspect, and how to compute them. It also follows cleanly from these properties that if one vector is in the span of the others, then the determinant is zero. This is also clear geometrically. This shows that if the corresponding matrix is not injective, the determinant must be zero. It also shows why column operations do not effect the determinant. Cramer's rule is also a very short computation from these properties.

I must admit, the other topics you mention (row operations, cofactor expansion, and classical adjoint) all seem somewhat tricky from this perspective.

Answer 6

Expansion by "cofactors" is really a form of "cancellation." That is, you try to get to a point where all the terms but one in a row or column is zero, which equation "trivializes" the exercise. Of course, when you add or subtract columns or rows to produce this result, you end up "compensating" the other terms in the array.

So the whole operation is one of "compensated" cancellation to determine the volume of the matrix.