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Before teaching the chapter on determinants in a linear-algebra course for beginning undergraduate students (mathematics and computer science, more specifically) I would like to give a small introduction and convince the students that this is an interesting notion.

What could one tell beginning undergraduate students in 15 to 30 minutes to get them more interested in determinants?

What then would follow are the "usual things" (special cases for small dimension, multi-linearity, computation via row and column transform, computation by developing along a line/column, Leibniz formula, Cramer's rule and formula for the inverse matrix).

I might modify this slightly if it helps to make the motivation make sense in retrospect, but to the extent possible I would like to keep the question focused on an introductory presentation (rather than the actual content), as I am aware of the question "Is there a good way to explain determinants in an elementary linear algebra class?" which covers the aspect how to teach determinants.

To high-light and to reiterate, the intended difference between the two questions is that while the referenced one asks about how to actually teach the subject I am mainly interested in advice for a short (non-technical) introduction to be given before the subject of determinants is actually discussed. Thus, I would be willing to do some hand-waving and over-simplifications to convey an application of determinants and the related ideas also in areas currently beyond the students reach.

Something I would like to say is: Knowing the determinant is important to doing ThisInterestingThing?

I am mainly looking forward to answers in this sense, but more general considerations are welcome, too.

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    $\begingroup$ In fact, I am teaching linear algebra this term and sooner or later I'll come to the same sort of question, hoping to find a personal answer for. Meanwhile, you might have a look at the answers given to this question, math.stackexchange.com/questions/235939/… $\endgroup$ – Amir Asghari Oct 13 '15 at 21:52
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    $\begingroup$ Something of which I have an interest, but little practical knowledge of, is D-optimal designs. A professional statistician or combinatorialist may be able to explain why these determinant-maximizing integer arrays have application; I find certain aspects of the range of the determinant function on certain finite sets of interest. In particular, so little is known, even for small orders and small ranges of entry values. There is quite a bit of low hanging fruit for those who want to do computer experiments. Gerhard "May Go Get An Apple" Paseman, 2015.10.13 $\endgroup$ – Gerhard Paseman Oct 13 '15 at 23:34
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    $\begingroup$ Usually I have the opposite problem. Students are too keen on determinants, and overuse them. $\endgroup$ – Federico Poloni Oct 14 '15 at 9:53
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    $\begingroup$ Don't. Determinants are a relic that should be ditched from the undergraduate curriculum. Down With Determinants! Generally, if solving your problem means calculating a determinant, there's almost always a quicker way. $\endgroup$ – Loop Space Oct 14 '15 at 16:11
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    $\begingroup$ This is a bit late, but congratulations on your first question! :) $\endgroup$ – Joel Reyes Noche Oct 19 '15 at 1:40
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I have found it motivates to explain the determinant as computing a volume. One can work through and convince for $2 \times 2$ and $3 \times 3$ matrices, and perhaps only hint at the $n \times n$ generalization, when $|\det(M)|$ is the volume of the $n$-dimensional parallelepiped spanned by the column vectors of $M$.


          Parallelepiped
          (Image from Wikipedia's Determinant article.)
It is an added benefit that shuffling the column vectors clearly does not change the parallelepiped's absolute volume. One can then talk about signed area and signed volume.

These volume determinants are heavily used in computer graphics, providing more motivation. For example, signed area leads to the shoelace formula for computing the area of a simple polygon.

It would be a challenge to find a 3D computer game that does not use the signed volume determinant, e.g., to detect whether an object/point $p$ is inside or outside another object. Objects are triangulated, so one is computing the signed volume of a tetrahedron determined by a surface triangle $T$ and point $p$, which is a corner ($\frac{1}{6}$-th) of the parallelepiped. In other words,

"Knowing the determinant is important to doing..." collision detection in 3D games.

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To give a brief list of interesting applications:

  • Volume obviously the lead application. It is not unreasonable to say determinants are volumes. Of course, they're more than that, their signed-volumes. For example, if $v,w$ are three dimensional vectors tangent to a plane and we want to decide which side of the plane a third vector $u$ is on then the sign of $det(u|v|w)$ tells us if $u$ is up or down. This idea persists in $n=2$ (it's CW verses CCW there) and $n>3$.
  • Cramer's Rule can be used in both numerical and symbolic systems of equations. For example, it lies at the heart of the mysterious variation of parameter formulas you see in the differential equations course.
  • Exterior Algebra with the determinant in hand you can define wedge products as particular multilinear maps with formulas given by determinants. This is strongly connected to the larger program of detecting linear dependence by vanishing determinants or, vanishing wedge products.
  • Super slick formulas there are too many to list. Slater determinants, the two-point formula for a line, the three point formula for a plane, a differential equation with a given solution set, the inverse of a matrix, triple product identity, Lagrange's identity,...

Anyway, so much of the mystique stems from defining the determinant via Laplace's expansion by minors. Instead, we should introduce it in its raw multilinear glory.

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Maybe begin by saying the determinant is a test of whether a matrix is singular.

anti Loop Space:
Determinants must be retained, since they are routinely used in branches of learning other than abstract algebra. Physics, engineering, even ODE and PDE.

I agree we should not calculate large determinants by hand. But after completing your course, a student should have no problem understanding if his future lecturer says: "by Cramer's Rule, we see that the solution is a rational function of the variables". [an example from my personal experience]

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    $\begingroup$ I like the idea of an anti-loop space ... something for my next project, perhaps. Anyway, any lecturer who says "by Cramer's Rule ..." without reminding the students what that rule is deserves all the teaching evaluations that they get. Read my answer on the linked question. I am not anti-determinants in their entirety. I am against them in a beginner course on linear algebra taught to students who might never meet them again. Teach them when they are needed, no earlier. $\endgroup$ – Loop Space Oct 14 '15 at 20:17
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One very practical area (ok, mathematically practical...) where determinants are used quite often is in multivariable statistics -- and I'm not talking about advanced mathematical statistics.

For example, lets say you have a random vector (i.e., its components are random variables), the determinant allows you to calculate a single number that is the multivariate equivalent of a "variance" for this vector. That this can be gainfully done is not something that is immediately obvious for random vectors. Specifically, if $\Sigma$ is the covariance matrix, then $|\Sigma|$ is analogous to a univariate variance. This is especially apparent if you compare the univariate and multivariate Gaussian densities:

$$f(x) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$

$$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^k|\boldsymbol\Sigma|}} \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu}) \right)$$

Another important use of determinants is when you want to compute functions of random variables. For example, if $\mathbf{x}$ is a continuous random vector, where $\mathbf{x}$ has pdf $f(\mathbf{x})$ and we want to calculate the density function of $A\mathbf{x}+\mathbf{b}$, where $A$ is a non-singluar matrix, it turns out that it is not as bad as you'd initially think:

$$A\mathbf{x} + \mathbf{b} \sim \frac{1}{|A|}f(A^{-1}[\mathbf{x}-\mathbf{b}])$$

The quantity $\frac{1}{|A|}$ is called the Jacobian and ensures that when we integrate the transformed density function, it still integrates to 1 (see the connection to volume again?).

Anyway, to the extent that students see statistics as "practical" or "interesting", the above may be motivating examples of the everyday use of determinants.

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Knowing the determinant of a matrix lets you:

  • Solve the corresponding set of linear equations.
  • Find the eigenvalues of the matrix. The eigenvalues correspond to the growth rates of the vector solutions as the matrix is repeatedly multiplied by itself.
  • Invert the matrix.
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    $\begingroup$ No, No, and No. No-one solves linear equations using determinants any more. The likelihood of the matrix being invertible is small, and the numerical stability is appalling. Finding eigenvalues for anything bigger than 4x4 involves solving a quintic - or worse - equation which is impossible. To invert a matrix, you use Gaussian Elimination (or a variant thereof) without ever touching the determinant. $\endgroup$ – Loop Space Oct 14 '15 at 20:13
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    $\begingroup$ @LoopSpace: "The likelihood of the matrix being invertible is small"? Under just about any probability distribution, the probability of a matrix being invertible is one, i.e. "almost everywhere." You have a point here but it was badly stated. I agree with your other points. $\endgroup$ – Rory Daulton Oct 15 '15 at 10:06
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    $\begingroup$ Really? I think you should recount. This time, include all matrices. I think you'll find that the overwhelming majority are not invertible. $\endgroup$ – Loop Space Oct 15 '15 at 16:45
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Warning: I am not a design specialist or professional: use the fuzzy description below to inspire research, not replace it.

In manufacturing and operations research (and likely some branches of statistics) one is interested in economical designs for testing. This is sort of like testing in batches 100 glasses of wine in 7 tests to find one bad glass; in industry often several different characteristics are desired to be learned by sampling a large population.

A design is an array, usually object or characteristic for each column, and each row is a grouping: each entry indicates a value that is related to testing the group including that object. For the wine example, 0 means don't test and 1 means do test. In some situations, a D-optimal design is a square array of values which maximizes information gathered from a sequence (or even partial sequence) of tests: numerically this is by maximizing the determinant.

Even outside the context of designs, there are some open problems that are computationally feasible, but still take many cycles to run. Determinants of 0-1 matrices are related to the Hadamard maximal determinant problem and the determinant spectrum problem, both of which are open and for which even for matrices of order less than 20 a lot is left to known. Combinatorial matrix theory could benefit from some well-organized use of computer instruction cycles on small examples.

Gerhard "Determinant Means Volume Of Information" Paseman, 2015.10.20

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  • $\begingroup$ Thanks a lot for providing this answer. I will delete the obsolete comment now...just kidding ;-) $\endgroup$ – quid Oct 21 '15 at 22:08

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