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I am taking a proof-based introductory course to Linear Algebra as an undergrad student of Mathematics and Computer Science. The author of my textbook (Friedberg's Linear Algebra, 4th Edition) says in the introduction to Chapter 4:

The determinant, which has played a prominent role in the theory of linear algebra, is a special scalar-valued function defined on the set of square matrices. Although it still has a place in the study of linear algebra and its applications, its role is less central than in former times.

He even sets up the chapter in such a way that you can skip going into detail and move on:

For the reader who prefers to treat determinants lightly, Section 4.4 contains the essential properties that are needed in later chapters.

Could anyone offer a didactic and simple explanation that refutes or asserts the author's statement?

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    $\begingroup$ Sheldon Axler's paper "Down with Determinants" seems a useful example of the the anti-determinant POV. (He later wrote an entire linear algebra text with that approach.) $\endgroup$ – Semiclassical Nov 4 '16 at 21:35
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    $\begingroup$ I think it's useful for didactic purposes. But in engineering praxis where large matrices appear, people never use determinants (they are either zero or not computable) and use various iterative methods instead. $\endgroup$ – Peter Franek Nov 4 '16 at 21:36
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    $\begingroup$ It is a useful theoretical tool, but fairly hopeless from a floating point numerical standpoint. Another, more extreme, example is the Jordan form of a matrix. It provides invaluable insight, but is essentially impossible to compute numerically. $\endgroup$ – copper.hat Nov 4 '16 at 22:50
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    $\begingroup$ IIRC Gilbert Strang says basically the same $\endgroup$ – leonbloy Nov 5 '16 at 23:08
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    $\begingroup$ @leonbloy Thanks for making me look for Strang's text on intro to Linear Algebra. What a contrast! Friedberg's text feels boring and exhausting, already half through the book and I am tired and feeling like I am going nowhere. On the other hand, read two pages of Strang's and I suddenly want to know how the novel ends, to put it some way. It feels motivating, candid, "feynmanesque", human. Hell, the guy even writes some sentences in first person. Also, it is closer to Computer Science. All in all the textbook I was looking for. Thanks. $\endgroup$ – David Cabrera Nov 6 '16 at 0:21

10 Answers 10

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Friedberg is not wrong, at least on a historical standpoint, as I am going to try to show it.

Determinants were discovered in the mid 18th century, and had a rapid spread. Their discoverer, Cramer, used them in his celebrated rule for the solution of a linear system (in terms of quotients of determinants). This generation of mathematicians has discovered properties of determinants that now, with our vision, we express in terms of matrices.

Cauchy has extended the use of determinants as explained in the very nice article by Hawkins referenced below :

  • around 1815, he discovered the multiplication rule (lines times columns) of two determinants. This is typical of a result that has been completely remodeled ; nowadays, this rule is for the multiplication of matrices, and determinants' multiplication is restated as the homomorphism rule $\det(A \times B)= \det(A)\det(B)$.

  • around 1825, he discovered eigenvalues "associated with a symmetric determinant" and established the important result that these eigenvalues are real ; this discovery has its roots in astronomy, in connection with Sturm, explaining the word "secular values" he attached to them: see for example this).

Matrices made a shy apparition in the mid-19th century (in England) ; "matrix" is a term coined by Sylvester see here. I strongly advise to take a look at his elegant style in his Collected Papers.

Together with his friend Cayley, they can rightly be named the founding fathers of linear algebra, with determinants as permanent reference. Here is a major quote of Sylvester:

"I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent".

The characteristic polynomial is expressed as the famous $\det(A-\lambda I)$, the "resultant", invented by Sylvester (giving a characteristic condition for two polynomials to have a common root), is a determinant as well, etc.

Let us repeat it : for a mid-19th century mathematician, a square array of numbers has necessarily a value (its determinant): it cannot have any other meaning. If it is a rectangular array, the numbers attached to it are the determinants of submatrices that can be "extracted" from the array.

The identification of "Linear Algebra" as a full (and new) part of Mathematics is mainly due to the German School (from 1870 till the 1930's with Van der Waerden's book : "Moderne Algebra" that is still very readable). I don't cite the names, there are too many of them. An example among many others of the german dominance: the germenglish word "eigenvalue". The word "kernel" could have remained the german word "kern" that appears around 1900 (see this site).

The triumph of Linear Algebra is rather recent (mid-20th century). "Triumph" meaning that now Linear Algebra has found a very central place. Determinants in all that ? Maybe the biggest blade in this swissknife, but not more ; another invariant (this term would deserve a long paragraph by itself), the trace, would be another blade, not the smallest.

A special treatment should be made to the connection between geometry and determinants, which has paved the way, here in particular, to linear algebra. Some cornerstones:

  • the development of projective geometry, in its analytical form, in the 1850s. This development has led in particular to the modern study of conic sections described by a matrix equation $X^TAX=0$ where $A$ is a symmetrical $3 \times 3$ matrix (no determinant can express the latter relationship). Different fundamental ideas were born from this representation. Three examples:

    a) the concept of rank: for example, a pair of straight lines is a conic section whose matrix has rank 1. The "rank" of a matrix used to be defined in an indirect way as the "dimension of the largest nonzero determinant that can be extracted from the matrix". Nowadays, the rank is defined in a more straightforward way as the dimension of the range space... at the cost of a little more abstraction.

    b) the concept of duality $X=(x,y,t)^T\rightarrow U=AX=(u,v,w)$ between points $(x,y)$ and lines with equations $ux+vy+w=0$. More precisely, the tangential description, i.e., the constraint on the coefficients $U^T=(u,v,w)$ of the tangent lines to the conical curve has been recognized as associated with $A^{-1}$ (under the condition $\det(A) \neq 0$!), due to relationship $X^TAX=X^TAA^{-1}AX=(AX)^T(A^{-1})(AX)=U^TA^{-1}U=0$. (see the excellent lecture notes : (http://www.maths.gla.ac.uk/wws/cabripages/conics/conics0.html)).

Remark: the way the latter equalities have been written use matrix algebra notations and rules that were unknown in the 19th century, with the notable exception of Grassmann's "Ausdehnungslehre", whose ideas were too ahead of his time (1844) to have any influence.

c) the concept of eigenvector/eigenvalue, initially motivated by the determination of "principal axes" of conics and quadrics.

  • the very idea of "geometric transformation" (more or less born with Klein circa 1870) which, when it is a linear transformation, is associated with an array of numbers. A matrix, of course, is much more that an array of numbers... But think for example to the persistence of the expression "table of direction cosines" (instead of "orthogonal matrix") as can be found for example still in the 2002 edition of Analytical Mechanics by A.I. Lorrie.

  • and quaternions, etc...

References:

I just discover, 3 years later, a rather similar question with a very complete answer by Denis Serre, a French specialist in the domain of matrices : https://mathoverflow.net/q/35988/88984

The article by Thomas Hawkins : "Cauchy and the spectral theory of matrices", Historia Mathematica 2, 1975, 1_29.

See also (http://www.mathunion.org/ICM/ICM1974.2/Main/icm1974.2.0561.0570.ocr.pdf)

An important bibliography is to be found in (http://www-groups.dcs.st-and.ac.uk/history/HistTopics/References/Matrices_and_determinants.html).

See also a good paper by Nicholas Higham : (http://eprints.ma.man.ac.uk/954/01/cay_syl_07.pdf)

For conic sections and projective geometry, see a) this excellent chapter of lectures of the University of Vienna (see the other chapters as well) : (https://www-m10.ma.tum.de/foswiki/pub/Lehre/WS0809/GeometrieKalkueleWS0809/ch10.pdf). See as well : (maths.gla.ac.uk/wws/cabripages/conics/conics0.html).

Don't miss the following very interesting paper about various kinds of useful determinants : https://arxiv.org/pdf/math/9902004.pdf

See also this

Interesting precisions on determinants in the answers here.

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    $\begingroup$ Very great answer. can you provide some sources to read more about the connections between determinants and geometry, specially conic sections? $\endgroup$ – Fawzy Hegab Nov 16 '16 at 7:34
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    $\begingroup$ @Fawzy Hegab Sorry to answer you so late, but, 6 months ago, I hadn't a good web reference to give you. Here is one:(maths.gla.ac.uk/wws/cabripages/conics/conics0.html) $\endgroup$ – Jean Marie May 5 '17 at 8:41
  • $\begingroup$ @JeanMarie I'm amazed how far I've come from this question in 2 years and how you still find ways to groom it and add new data to it. Kudos to you! :) $\endgroup$ – David Cabrera Mar 22 at 13:54
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It depends who you speak to.

  • In numerical mathematics, where people actually have to compute things on a computer, it is largely recognized that determinants are useless. Indeed, in order to compute determinants, either you use the Laplace recursive rule ("violence on minors"), which costs $O(n!)$ and is infeasible already for very small values of $n$, or you go through a triangular decomposition (Gaussian elimination), which by itself already tells you everything you needed to know in the first place. Moreover, for most reasonably-sized matrices containing floating-point numbers, determinants overflow or underflow (try $\det \frac{1}{10} I_{350\times 350}$, for instance). To put another nail on the coffin, computing eigenvalues by finding the roots of $\det(A-xI)$ is hopelessly unstable. In short: in numerical computing, whatever you want to do with determinants, there is a better way to do it without using them.
  • In pure mathematics, where people are perfectly fine knowing that an explicit formula exists, all the examples are $3\times 3$ anyway and people make computations by hand, determinants are invaluable. If one uses Gaussian elimination instead, all those divisions complicate computations horribly: one needs to take different paths whether things are zero or not, so when computing symbolically one gets lost in a myriad of cases. The great thing about determinants is that they give you an explicit polynomial formula to tell when a matrix is invertible or not: this is extremely useful in proofs, and allows for lots of elegant arguments. For instance, try proving this fact without determinants: given $A,B\in\mathbb{R}^{n\times n}$, if $A+Bx$ is singular for $n+1$ distinct real values of $x$, then it is singular for all values of $x$. This is the kind of things you need in proofs, and determinants are a priceless tool. Who cares if the explicit formula has a exponential number of terms: they have a very nice structure, with lots of neat combinatorial interpretations.
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    $\begingroup$ I'm not sure that the pure/applied dichotomy is accurate... maybe "numerical" versus "non-numerical"? E.g., not all numerically oriented math is applicable, and ... etc. $\endgroup$ – paul garrett Nov 4 '16 at 22:37
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    $\begingroup$ There ios an exception to this rule: In some contexts, what you want to compute is the determinant of some matrix (like in D-optimal experimental design!) then there is no shortcut to actually compute it (but numerically, it makes then more sense to compute the logdeterminant). $\endgroup$ – kjetil b halvorsen Nov 5 '16 at 14:18
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    $\begingroup$ Since in D-optimal design you want the determinant of a matrix like $X'X$, the quickest and most numerically stable way to find it is from the singular values of $X$, not by multiplying out $X'X$ and then finding its determinant by an elementary method. Of course calculating the log of the determinant from the singular values is just as trivial as calculating the determinant itself. $\endgroup$ – alephzero Nov 8 '16 at 4:26
  • $\begingroup$ $X^TX$ is a bit like determinants -- whatever numerical computation you want to do with it, there's usually a better way to do it that avoids it overall. And the same holds for matrix inverses. $\endgroup$ – Federico Poloni Jun 2 '18 at 13:28
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Determinants are still very much relevant to abstract algebra. In applied mathematics, they are less so, though the claim that "determinants are impractical because they take too long to compute" is misguided (no one forces you to compute them by the Leibniz formula; Gaussian elimination works in $O\left(n^3\right)$ time, and there is an $O\left(n^4\right)$ division-free algorithm as well). (Another oft-repeated assertion is that Cramer's rule is not very useful for solving actual systems of linear equations; I believe this one is correct, but I am mostly seeing Cramer's rule used as a theoretical tool in proofs as opposed to computational applications.)

What is going on is the following: Back up to the early 20th century, determinants used to be one of the few tools available for linear-algebraic problems. (They might even be one of the oldest tools, discovered by Takakazu Seki back in 1683.) Other linear-algebraic tools started appearing in the 18th and 19th centuries, but their development had always been lagging behind that of determinants until the likes of Noether and Bourbaki came around in the 20th century. (Muir's 5-volume annals of determinant theory, which can be downloaded from Alain Lascoux's website, contain an impressive collection of results, many of them deep, about determinants.) Thus, for a long time, the only way to a deep result would pass through the land of determinants, simply because other lands were barely explored. Only after the notions of vector spaces, modules, tensors, exterior powers etc. went mainstream (1950s?), mathematicians could afford avoiding determinants, and they started noticing that it would often be easier to do so, and with the benefit of hindsight, some of the older uses of determinants were just detours. At some point, avoiding determinants became something like a cultural fashion, and Axler took it to an extreme in his LADR textbook, emphasizing non-constructive methods and leaving students rather ill-prepared for research in abstract algebra. Nevertheless, Axler's approach has some strengths (e.g., his nice and slick determinant-free proof of the existence of eigenvectors has become standard now, and is included even in Treil's LADW book, whose title is a quip on Axler's), which once again illustrates what I think is the correct takeaway from the whole story: Determinants used to be treated as a panacea, for lack of other tools of comparable strength; but now that the rest of linear algebra has caught up, they have retreated to the grounds where they belong, which is still a wide swath of the mathematical landscape (many parts of abstract algebra and algebraic combinatorics have determinants written into their DNA, rather than using them as a tool; they are not very likely to shed them off).

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    $\begingroup$ +1 for pointing out that the evaluation is not the limiting factor with fp. computations. $\endgroup$ – copper.hat Nov 4 '16 at 22:54
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Determinants are very useful in the theory of linear algebra, but less so in practice (especially for numerical computations involving large systems). For example,

  1. Cramer's rule gives an explicit formula for solving an $n \times n$ linear system $Ax = b$ where $A$ is invertible.
  2. The eigenvalues of a square matrix $A$ are the roots of the characteristic polynomial $\det(A - \lambda I)$.

But nobody uses these for computations, except for very small $n$. There are numerical methods that are much more efficient and numerically stable.

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Apart from the points already raised that determinants are in fact expensive to compute numerically and sensitive to rounding errors, they are also awkward "symbolically"... and indeed capture just a very peculiar bit of the linear algebra going on even in solving systems of linear equations.

The issue that does seem to disturb people, pro-and-con, if they are thinking in terms of "correct logical development" of linear algebra (which I myself think is already a viewpoint that's asking for trouble), a genuine and legitimate objection is over-use of the Cayley-Hamilton theorem... E.g., it is not necessary to use this in most of the situations where it is traditionally invoked.

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The determinant is relatively simple to calculate (computationally speaking) and has a number of interesting interpretations/applications attached to it, including:

  • Determining whether or not a matrix is invertible
  • Indicating by how much the transformation changes volume (and this is also related to differentiable transformations via the Jacobian; see change of variables)
  • Counting things! Determinants have lots of use in combinatorics (for instance, Kasteleyn matrices)
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  • $\begingroup$ I knew about the first two applications, and I sincerely appreciate the answer. But I am more interested in understanding why Friedberg treated determinants with certain air of obsolescence. As if they were no longer as useful or important as they used to be. I am not trying to understand why they are important, but to understand why Friedberg presented them as a thing of the past. $\endgroup$ – David Cabrera Nov 4 '16 at 21:07
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    $\begingroup$ Well, I can only say I (like your professor) disagree with Friedberg. $\endgroup$ – Fimpellizieri Nov 4 '16 at 21:12
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    $\begingroup$ I understand. See, mathematical concepts are usually treated with canonical respect, therefore it is just reasonably intriguing to read that statement. Friedberg must be, to a reasonable extent, a reputable source. I expect that he would not make such assertion without an explanation, whether it is a good one or not. I am just intrigued, that's it. $\endgroup$ – David Cabrera Nov 4 '16 at 21:20
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    $\begingroup$ "The determinant is relatively simple to calculate" is a gross misrepresentation of reality. Computing determinants is very computationally demanding and extremely sensitive to rounding errors. This is part of the reason Friedberg makes his claim, and he is without a doubt correct. $\endgroup$ – Ittay Weiss Nov 4 '16 at 21:46
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    $\begingroup$ @Fimpellizieri $O(n^3)$, for practical computations, is quite demanding. And as you say deciding whether the determinant is zero, which is commonly required in computations, tends to be ill-posed. That, together with the existence of quite efficient iterative methods which tend to be computationally robust, is probably at the heart of the comment which led to the question. $\endgroup$ – Ittay Weiss Nov 4 '16 at 23:58
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A complete outsider perspective: the twentieth century has seen an increasing generality and abstraction in mathematics. Thus in this thrust things like determinants (or even matrices) can be seen as overly restrictive or burdensome as they cannot describe all infinite dimensional spaces and their basis in calculation can be used to obfuscate general concepts.

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  • $\begingroup$ Hopefully someone more experienced and knowledgeable can comment on my answer so I might learn something. $\endgroup$ – Jacob Wakem Nov 5 '16 at 20:59
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    $\begingroup$ Just because determinants can't handle all situations in infinite-dimensional spaces where they were used for the same problems in finite-dimensional spaces doesn't mean they can't be useful for some infinite-dimensional problems. Dwork's solution in the 1960s of the first part of the Weil conjectures (rationality of the zeta-function of algebraic varieties over finite fields) relies on infinite-dimensional determinants of operators on $p$-adic Banach spaces. $\endgroup$ – KCd Nov 5 '16 at 23:19
  • $\begingroup$ @KCd There is no such thing as an infinite-dimensional determinant because a determinant is a number. $\endgroup$ – Jacob Wakem Oct 23 '18 at 18:32
  • $\begingroup$ I have no idea what you mean. Infinite series, infinite products, and determinants of infinite-dimensional matrices are all standard concepts in analysis (when they are convergent). An infinite series like $1 + 1/2^3 + 1/3^3 + 1/4^3 + \cdots$ is a number. Are you going to tell me there is no such thing as an infinite series too? $\endgroup$ – KCd Oct 23 '18 at 19:43
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To me it seems the author lacks a sufficient comprehensive knowledge of the range of applications of determinants and so the complete statement should be taken with care.

A field of application is combinatorics and there are surprising interpretations of determinants which also contribute to their power and usefulness.

An example is the following interpretation:

[M. Fulmek (2012)]: $(m\times m)$-determinants may be viewed as (generating functions of) $m$-tuples of nonintersecting lattics paths in $\mathbb{Z}^2$ with starting points on some fixed horizontal line $y=\alpha$ and ending points on some fixed horizontal line $y=\omega>\alpha$, where

  • the rows of the determinants (in the usual order: top to bottom) correspond to the starting points of the lattice pathes (ordered from right to left),

  • and the columns of the determinants (in the usual order: left to right) correspond to the ending points of the lattice pathes (ordered from right to left).

which is from his paper Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively.

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  • $\begingroup$ Thanks for the excellent reference [Fulmek], that later on I found as well in(en.wikipedia.org/wiki/…). It is a typical case where determinants are an essential tool. I will make a comparison between determinants and trigonometry : if you have been overfed with determinants or trigonometry in your student's years, a natural trend is to say "too much is too much" : but in one case as the other, you need it. For example, you couldn't do much geometry without trigonometry... $\endgroup$ – Jean Marie Dec 4 '17 at 7:49
  • $\begingroup$ The reason why I am interested by these questions is that I am presently working on the fascinating Robinson-Schensted-(Knuth) correspondence (see (cmi.ac.in/~ksutar/reptheory/Viennot.pdf)) $\endgroup$ – Jean Marie Dec 4 '17 at 7:55
  • $\begingroup$ @JeanMarie: You're welcome, and many thanks for this interesting reference, I appreciate it. :-) $\endgroup$ – Markus Scheuer Dec 4 '17 at 18:32
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    $\begingroup$ Yes, I fully agree with you (and Christian Blatter) that, from a pedagogocal point of view, it is a nonsense to prone the exclusion of determinants. Students, through application of simple algorithms, be it the computation of $det(A-\lambda I)$, or Euclid's algorithm, or the computation of roots of a quadratic, or... learn a lot and get acquainted with the corresponding theme (and - very important - are happy with it); on this basis, one can build more abstract concepts. If you feed them all the way long with abstract concepts, you will loose a big part of your audience...(ctd) $\endgroup$ – Jean Marie Dec 4 '17 at 21:15
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    $\begingroup$ ...(ctd) (I think here to undergraduate students ; the issues are not the same later : people have gained confidence into themselves, and are much more apt to abstraction) $\endgroup$ – Jean Marie Dec 4 '17 at 21:17
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Back when I learned Linear Algebra, the text book we used found the "inverse matrix" using the a combination of the "minors", divided by the determinant.

Today it is much more common to use "row reduction" to find an inverse matrix or solve a system of equations.

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    $\begingroup$ @Ittayweiss I strongly discourage you to tell people to delete their answers. Rather, they can be encouraged to improve their answer. $\endgroup$ – Pedro Tamaroff Nov 5 '16 at 1:43
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    $\begingroup$ Thank you Pedro for your comment. To answer poster: please note that your answer reads more like a comment rather than an actual answer. It only very loosely addresses the question OP articulated. Please consider modifying your answer so that it more comprehensively answers the question, or place it as a comment. $\endgroup$ – Ittay Weiss Nov 5 '16 at 1:54
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Throw this book away!

It may be that certain sub-sub-fields of algebra that make heavy use of determinants as symmetric polynomials are not so much en vogue now as they were a hundred years ago. Such things happen, no bad feelings about that. But it is beyond me how an author of a textbook directed at students of all kinds of mathematics can seriously state that the importance of the notion of determinant "is less central than it was in former times". Such an utterance is similar to the statement that "the real numbers are less central than they were in former times" from an author of a textbook about discrete mathematics.

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    $\begingroup$ With uttermost respect sir, your indignation may be reasonable but I would appreciate it if you go into further detail. Maybe I lack the formal understanding, remember that I am just an undergrad student. Thanks anyways. $\endgroup$ – David Cabrera Nov 4 '16 at 21:36
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    $\begingroup$ Suggesting throwing away books based on one sentence. That is really good advice. Lets instead throw away answers who only provide arguments on the form "because I say so". $\endgroup$ – Winther Nov 4 '16 at 22:00
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    $\begingroup$ Actually, Christian Blatter has a good point. The key word is "students." In the 60's academia started, and has now finished, selling itself down the river for the sake of job training. All universities are now DeVry or ITT Tech. We answer questions only based on "when will this be used". Never on the basis of "does this improve a student's insight." It's one argument whether determinants are useful in calculations of any sort. It's quite another whether they're useful pedagogically. And since it's a _text_book, the pedagogical consideration should be paramount. $\endgroup$ – B. Goddard Nov 4 '16 at 23:01
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    $\begingroup$ I suspect that Christian is using the term in a conversational rather than a book burning sense. The back of the book states "For statisticians and engineers.", and, in this context, it is true that determinants are not terribly relevant. $\endgroup$ – copper.hat Nov 4 '16 at 23:19
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    $\begingroup$ The real numbers are less central than they were in former times :) I think there has been a movement away from "try to do all of mathematics in just a few big sets ($\mathbb{R}$, $\mathbb{C}$, $L^2\left(\mathbb{R}\right)$, etc.)" towards "try to do each thing in whatever place it feels most natural" (e.g., computations with polynomials are done over arbitrary commutative rings; Galois theory has moved from number fields into arbitrary field extensions; certain results went all the way up to abelian categories, etc.). In this sense, almost everything has become less central! $\endgroup$ – darij grinberg Nov 4 '16 at 23:56

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