# Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better conceptualization of the determinant.

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+1 good question. You're in for a wilder ride than you probably thought. Determinants were arguably the first linear algebra concept to be invented -- before abstract vector spaces, before even matrices. There were just these strange expressions that kept popping up when quite different problems were solved the hard way. Sometime along the way someone noticed that they were all structured similarly and gave them a name. I hope you get some good historical references in answers. – Henning Makholm Nov 13 '11 at 2:32
Non-duplicate: where did determinant come from? which was closed as a duplicate of a non-historical intuition-of-determinants question. One of the answers before it was closed links to an interesting historical wall-of-text, though. – Henning Makholm Nov 13 '11 at 2:37
If you are looking for something involving the history of determinants (and matrices) you might start here: www-history.mcs.st-and.ac.uk/history/HistTopics/… – Joseph Malkevitch Nov 13 '11 at 15:33
I know of two very interesting things; A chinese mathematician (B.C.) recorded the determinant of a 3x3 system; see books.google.co.uk/books/about/… and also Euler or Lagrange in a litter wrote down a system of equations in a letter as 11 + 12 + ... +17 = x (say) then 21 + 22 + ... + 27 = y and ... and ...81 + 82 + ... + 87 = z where 11 does not represent 11 but the first sum of the first equation - in matrix notation, if the system was represented by a matrix A cont. – Adam Jan 25 '12 at 8:51
11 would be writte a_1,1 and in general ij is a_ij. He wrote down a condition like 11.22.33 + ... - 12.34.25 = 0 for the system to have a solution that's non-trivial, that is the determinant. – Adam Jan 25 '12 at 8:53

Let me tell you what I know in general.

The determinant was primarily introduced as a gauge to measure the existence of unique solutions to linear equations. It's like a litmus paper (which is used to know about acids and bases, but in this case its the existence of unique solutions). If you doubt that one can measure the uniqueness of solutions, I have a pair of magic spectacles you may keep that enable you to visualize the determinant in a new geometric way (this in fact can be found in many standard books, but I wanted to write the quintessence here).

In response to your first question about origin, it dates back to the $3^{rd}$ century, when Chinese mathematicians used determinants in their book by name The Nine Chapters on the Mathematical Art (Chinese version and English version here ). At first, when the historians used the concept of determinants, they didn't refer to matrices but merely to the system of linear equations and treated the determinant as a property which tests for the existence of unique solutions for a system of linear equations. And later, due to the discovery of matrix theory, determinants have been moved to the theory of matrices (in a new manner of evolution).

If you consider a matrix $$\begin{pmatrix} a&b\\c&d \end{pmatrix}$$ the determinant is given by $ad-bc$. You may wonder what information does $ad-bc$ may have within itself. All you need to do is to view things in geometric manner when you are unable to visualize things in algebraic way. Matrices are closely related to fields: in vectors and linear algebra, we representing vectors in terms of matrices. They can be seen as collections of vectors. The example that I have in mind is: consider a billiard table. Before hitting the balls, you take a thing which has a triangular frame and then arrange all the balls (present in an irregular manner) into a triangular shape. It's like arranging things in an organised manner, as human beings always try to do.

Let me explain the topic in greater detail.

If you look at the above figure, you can clearly see the coordinates. You can think of the columns of the matrix as vectors and the entities in the column representing their Cartesian placement (the coordinate positions). So now if you take all the above things into consideration, you can clearly see that if $ad-bc=0$ then the rhombus suddenly became a straight line, and in generalized manner a parallelopiped suffers a decrement in dimension by one (making the volume zero, which indirectly implies that it has an area). So analogously one can see that if the determinant becomes zero, according to the Cramer's rule, it makes the denominator go $\infty$, which is a violation.

Some of the important alternative notions about determinants: (stress on second one)

• Determinants in reality can be thought as a measure of multiplicative change in the volume of parallelopiped when it is subjected to linear transformation. It can be shown as :

• And the main notion that answers the connection between the determinant and existence of solutions is that the determinant of a matrix is zero if and only if the column vectors of the matrix are linearly dependent. Thus, determinants can be used to characterize linearly dependent vectors. For example, given two vectors $v_1, v_2$ in $R^3$, a third vector $v_3$ lies in the plane spanned by the former two vectors exactly if the determinant of the $3$-by-$3$ matrix consisting of the three vectors is zero. So one needs to take the theory of multilinear forms into consideration. I can't express the entire theory, but to give you a short notion, the determinant is actually a multi-linear form in general. So in a deep sense it measures the manifestations of the things related to vectors.

• And when determinants are negative, they have a role of orientation in geometric sense, which is another crucial point.

Some Beautiful References :

I have some suggestions for you. Apart from reading about matrix theory in Wikipedia page I suggest reading a very good article Making Determinants Less Weird by John Duggan. And I have come across another good article recently, which is here

P.S : I took much time to edit and post this, so users are kindly requested to post any suggestions, in case of down-votes.

Thank you.

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Is it sufficient or I need to add something ? – Iyengar Feb 10 '12 at 16:00
"To your astonishment,..." seems rather presumptuous. You know how you felt, do you really feel the need to tell people how to feel about something which may or may not be a revelation to them? – Arturo Magidin Feb 10 '12 at 17:12
@ArturoMagidin : Ok, sorry sir, I have fixed it. Thank you. – Iyengar Feb 10 '12 at 17:13
Hello! I've noticed that your link to "Making Determinants Less Weird" has broken - do you know an alternate location this resource may be found? – Emrakul Oct 17 '14 at 4:39
Sorry @Emrakul, I am not active on M.SE anymore. But a simple Google search, would have done. Anyway the link's here : dl.dropboxusercontent.com/u/17516137/RapidWeaverSite/resources/… – Iyengar Dec 5 '14 at 9:13

The authoritative reference seems to be the books by Muir: Contributions to the history of determinants and A treatise on the theory of determinants. See also http://www-history.mcs.st-and.ac.uk/Extras/Muir_determinants.html.

There is also Miller, On the History of Determinants. Amer. Math. Monthly 37 (1930), no. 5, 216–219.

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