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?
Answer
Friedberg is not wrong, at least on a historical standpoint, as I am going to try to show it.
Determinants were discovered in the second half of the 18th century, and had a rather rapid spread. Their discoverer, Cramer, used them in his celebrated rule for the solution of a linear system (in terms of quotients of determinants). Mathematicians of the next two generations discovered properties of determinants that now, with our vision, we mostly 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 quadratic form, that can be written under an all-matricial expression $X^TMX=0$ where $M$ is a symmetrical $3 \times 3$ matrix
Different examples of the emergence of new trends :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 linear transformations and duality arising from geometry: $X=(x,y,t)^T\rightarrow U=MX=(u,v,w)$ 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 $M^{-1}$ (under the condition $\det(M) \neq 0$!), due to relationship
$$X^TMX=X^TMM^{-1}MX=(MX)^T(M^{-1})(MX)=U^TM^{-1}U=0$$
$$=\begin{pmatrix}u&v&w\end{pmatrix}\begin{pmatrix}A & B & D \\ B & C & E \\ D & E & F \end{pmatrix}\begin{pmatrix}u \\ v \\ w \end{pmatrix}=0$$
whereas, in XIXth century, it was classical to write the previous quadratic form as :
$$U^TM^{-1}U=\begin{vmatrix}a&b&d&u\\b&c&e&v\\d&e&f&w\\u&v&w&0\end{vmatrix}=0$$
as a "bordered determinant" directly with matrix $M$.
(see the excellent lecture notes (http://www.maths.gla.ac.uk/wws/cabripages/conics/conics0.html)). It is to be said that the idea of linear transformations, especially orthogonal transformations arose even earlier in the framework of the theory of numbers (quadratic representations).
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.
A fundamental book on "The Theory of Determinants" in 4 volumes has been written by Thomas Muir : http://igm.univ-mlv.fr/~al/Classiques/Muir/History_5/VOLUME5_TEXT.PDF (years 1906, 1911, 1922, 1923) for the last volumes or, for all of them https://ia800201.us.archive.org/17/items/theoryofdetermin01muiruoft/theoryofdetermin01muiruoft.pdf. It is very interesting to take random pages and see how the determinant-mania has been important, especially in the second half of the XIXth century, with results of uneven quality. Matrices appear at some places with the double bar convention that lasted a very long time. Matrices are mentionned here and there, rarely to their advantage...
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