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Where did matrices and determinants come from?

By Murray Bourne, 08 Apr 2008

A reader of the Matrices and Determinants chapter in Interactive Mathematics recently wrote and asked where matrices and determinants come from and why do they work?

Matrices are essential for solving large sets of simultaneous equations using a computer. We certainly don't want to use a different letter for each variable in our problem (or lots of subscripts, like a34) because it would slow down the solution process and would be horrible to code. With matrices, we don't have to include any variables - just the numbers in front of those variables.

So for example, suppose we are trying to solve this 4x4 system of equations:

3x + 4y + 2z −6w = 5
x − 5y + 7z + 10w = −8
8x + 5yz + 7w = 8
6x − 4y + 12z + 15w = 4

We only need to give the computer the coefficients, like this:

3   4   2  −6  |  5
1  −5   7  10  | −8
8   5  −1   7  |  8
6  −4  12  15  |  4

The computer just works on the numbers − it doesn't need the letters.

The Han Chinese and Simultaneous Equations

Here's a problem from a Chinese mathematics book written in 200BC. (Source)

There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type?

It looks a lot like the kind of problems in textbooks today, doesn't it?

The remarkable thing about this problem is the way that the Chinese writer solved it. First, they set up the numbers involved as follows:

 1   2   3
 2   3   2
 3   1   1
26  34  39

(They are using rows where we would use columns. It doesn't matter.)

The instruction is to...

...multiply the middle column by 3 and subtract the right column as many times as possible, the same is then done subtracting the right column as many times as possible from 3 times the first column. This gives

 0   0   3
 4   5   2
 8   1   1
39  24  39

A similar process occurs to eliminate the 4 in the second row.

 0   0   3
 0   5   2
36   1   1
99  24  39

From this, we can read off the answer for the amount of the 3rd type (99/36 = 11/4) and then substitute to find the second type (17/4) and first type of corn (37/4).

We now call this process Gaussian Elimination after the German mathematician Gauss (1777-1855).

Maybe it should be called Han Elimination.

You can read more interesting history about matrices and determinants from The MacTutor History of Mathematics archive.

See the 12 Comments below.

12 Comments on “Where did matrices and determinants come from?”

  1. peter a. musa says:

    You have actually given some of us a good background in mathematics. With one's good knowledge of the origin of matrices and determinants, one would not feel the pain of solving problems in certain aspects of maths anymore.
    Thanks.

  2. Murray says:

    Thanks for the comment, Peter.

    Many people feel great pain solving matrices problems - they are tedious and prone to many errors.

    Let's use computers for this hack work and spend the time understanding what it means!

  3. Manos says:

    Thanks for this post. This gave a bit of motivasion on studying about Linear Algebra 🙂

  4. Brian says:

    It is many years since I tried advanced Maths and I never got the hang of why use this strange, unintuitive way of representing algebraic expressions. I don't pretend to be talented in maths although I think I was very badly taught in this area - I suspect the teacher hadn't got much of a clue.
    However, the penny may be starting to drop with your explanation.
    This is how I am now seeing it:
    1- you can solve a group of simultaneous equations using the coefficients in a specific way.
    2 - with a large number of 'dimensions' and/or a large number of equations, the 'workings out' (while still appearing complex and confusing) are vastly reduced in complexity and confusion compared to sticking with basic algebraic 'workings out'.

    Thanks for the information. I will keep an eye on this site.

  5. Murray says:

    @Brian. I'm glad you found the article useful. Your 2-point summary is a good one!

  6. Diego says:

    Hello,

    I had had this kind of explanation before, but I would like, if possible, to understand what does it mean (a matrix) in normal three dimensional space. I mean maybe I could see that 3x+2=y and 2x+1=y are two lines in a plane and that solving them means finding where those lines meet, but is there any kind of trick to be able to picture the solutions of a bigger and more complex group of equations??. Thank you very much in advance! 🙂

  7. Murray says:

    @Diego: For a 3x3 system, the solutions are represented by intersecting planes. If there is one unique solution, then the 3 planes represented by the 3 equations in 3 unknowns meet at a point.

    For larger systems, it's a bit trickier to visualize.

    These pages may help:
    MathWarehouse
    Wolfram Demonstration (requires the CDF plugin, but well worth it)

  8. Gaurav Bhattacharjee says:

    Well, apparently, this question took birth in my mind too. I'm simply fascinated by matrices. I heard of a branch of quantum mechanics called "matrix mechanics". Can you please be kind enough to tell me something about this study?

  9. Murray says:

    @Gaurev: This topic is beyond the scope of IntMath, but this search brings up plenty of results.

  10. Freemon Sandlewould says:

    You can see the rest of the story on matrix determinants here:http://www.amarketplaceofideas.com/math-derivation-of-matrix-determinant.htm

  11. kevin thomas says:

    I am a person who has been terrorized by math since the 5th. grade with the era of "the new math." I learned nothing in school about math, nothing. I have since learned math on my own and love it and have found your tutorials excellent without exception.

    The problem I have is that this Han Chinese method/example you outline went by me like I was frozen. You give a two sentence description which seemed to leave out something(s) important. I followed what came before in the chapter with no problem, but the logic of the description for this thing read like, well like Han Chinese to me. It made no sense, again, to me. I followed your instructions and got different answers (except in the first step, I think). I read your instructions at least 6 times and ended up even more confused.

    This points to a huge issue I have always had with math, and is really the reason I'm writing at all...word problems! The thing with word problems is that they can be imprecise and confusing, like you know...words, and can be read and understood in different ways; on timed exams this translates to panic and brain death, as I learned over and over.

    Roger Penrose tells a very instructive story about his difficulties in math when he was young, in the very beginning chapter of his book "Road to Reality." Like him I was not a student who could work math tests fast (and speed is really what's being tested, as Penrose pointed out, not comprehension) only slowly. That's the real problem with so many students who turn their backs on math. It is the velocity at which teachers try and cram information into you that's far too high (for most) and, critically, if you fall behind...you die! You probably already know what I myself noted in school, student textbooks have virtually no solved problems and very little on technique. I would guess that this is supposed to be some kind of exhortation for students to try hard and/or not cheat. It didn't work for me nor for all who hate math now. The logic, I think, assumes that the student will have, at some point, some kind of "come to Jesus" moment, and it'll all make sense. A moment that that for most never comes. The logic of how math is taught implies that in order to learn it you already need to know it. Why else keep the answers a secret and NOT give tons of examples? Are teachers trying to not teach math or what?

    I see math as both beautiful AND utilitarian. If most students don't grasp the beauty in math (at least at first), they should at least be allowed to try and grasp it's utility. I learned math only slowly, with tons of examples, hard work and, most importantly, it was required in my job; not by divine inspiration or natural talent as I possess neither. Put examples and answers in textbooks, please.

    Sorry, but I had to get that off my chest.

    kt

  12. Murray says:

    @KT: Sorry the explanation was somewhat "thin". Let me try to rectify that.

    This was the starting point of the example given:

    Start
    1 2 3
    2 3 2
    3 1 1
    26 34 39

    The above actually represents the following system of three equations in three unknowns (we are going down the columns).

    x + 2y + 3z = 26
    2x + 3y + z = 34
    3x + 2y + z = 39

    The first step is to multiply the second column of the above by 3, which gives us the new (green) column 2. The other columns are unchanged.

    We continue as per the instructions, subtracting Column 3 from Column 2 over and over, giving new green Column 2s.

    Col2 × 3
    1 6 3
    2 9 2
    3 3 1
    26 102 39
    Col2 − Col3
    1 3 3
    2 7 2
    3 2 1
    26 63 39
    Col2 − Col3
    1 0 3
    2 5 2
    3 1 1
    26 24 39

    Now, we multiply column 1 by 3 (since there is a 3 in the 3rd column), so we can obtain a zero in the first row, first column.

    Col1 × 3
    3 0 3
    6 5 2
    9 1 1
    78 24 39

    This next one is the first result given in the article above. We have 0 in 2 of the 3 cells in row 1.

    Col1 − Col3
    0 0 3
    4 5 2
    8 1 1
    39 24 39

    We continue on, but this time we need to multiply column 1 by 5, since there is a 5 in the middle column, second row. (There are other ways we can go about it, but this will give us a useful "triangle of zeros" in the final step.)

    Col1 × 5
    0 0 3
    20 5 2
    40 1 1
    195 24 39
    Col1 − Col2
    0 0 3
    15 5 2
    39 1 1
    171 24 39
    Col1 − Col2
    0 0 3
    10 5 2
    38 1 1
    147 24 39
    Col1 − Col2
    0 0 3
    5 5 2
    37 1 1
    123 24 39

    This is the final step we need to do. We now have 2 zeros in the first column and one zero in the second column.

    Col1 − Col2
    0 0 3
    0 5 2
    36 1 1
    99 24 39

    It is now possible to solve the system, which actually now means (going down the columns):

    36z = 99

    5y + z = 24

    3x + 2y + z = 39

    The first row gives z = 99/36 = 11/4, the second row gives y = 17/4 and the thrid row gives x = 37/4.

    Hope it helps.

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