# 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 I hope you get some good historical references in answers.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.

Determinant was primarily introduced as a gauge to measure the existence of unique solutions to linear equations. Its like a Litmus paper ( which is used to know about acids and bases, but in this case its the existence of unique solutions). You may get a doubt that how can one measure the uniqueness of solutions, for that I have a pair of magic spectacles, you need to keep them, so that you can visualize the determinant in a new geometric way ( This in-fact can be found in many standard books, but I want to write the quintessence here ) .

In correspondence to your first question, about origin, it dates back to $3^{rd}$ century, when Chinese mathematicians, where they used the determinants in their book by name The Nine Chapters on the Mathematical Art ( Chinese version and English version is 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 it as a property which measures existence of unique solutions for a system of linear equations. And later its due to the discovery of matrix theory , the determinants have been moved to the theory of matrices ( in a new manner of evolution ).

Suppose if you consider a Matrix, $$\begin{pmatrix} a&b\\c&d \end{pmatrix}$$ the determinant is given by $ad-bc$. So you may get a doubt that what information does $ad-bc$ has within itself. Just you need to do is to view things in geometric manner, when you are unable to visualize things in algebraic way. So you know that matrices are closely related to the fields : vectors and linear algebra, we are representing the vectors in terms of matrices. It can be seen as a collection of Vectors. The example that I got in my mind is that, consider a billiard table, before hitting the balls, you will take a thing which is triangular frame and then arrange all balls ( present in irregular manner ) into a triangular shape. Its like arranging things in an organised manner, and that is what human being always tries for.

So let me give the explanation about the topic in further .

If you look at the above figure, you can clearly see the coordinates, you can think the columns of the matrix, to be a vectors and the entities in the column representing their cartesian placement ( the coordinate positions ) . So now if you take all the above things into a consideration, now 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 to be 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 violation.

Some of the important alternative Notions about the 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 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 the determinants become 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 the theory of matrix which is in wikipedia page I will suggest you to read a very good article Making Determinants Less Weird by John Duggan. And I have come across some 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