Matrices and Determinants

Matrices and Determinants - Matrices and Determinants...

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Matrices and Determinants Traces back to 4 th Century BC More obvious in 2 nd Century BC The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains: There are two fields whose total area is 1800 square yards. One produces grain at the rate of 2 / 3 of a bushel per square yard while the other produces grain at the rate of 1 / 2 a bushel per square yard. If the total yield is 1100 bushels, what is the size of each field? The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. Indeed it is fair to say that the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. First a problem is set up which is similar to the Babylonian example given above:- 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? Now the author does something quite remarkable. He sets up the coefficients of the system of three linear equations in three unknowns as a table on a 'counting board'. 1 2 3 2 3 2 3 1 1 26 34 39 Cardan , in Ars Magna (1545), gives a rule for solving a system of two linear equations which he calls regula de modo and which [7 ] calls mother of rules ! This rule gives what essentially is Cramer 's rule for solving a 2 2 system although Cardan does not make the final step. Cardan therefore does not reach the definition of a determinant but, with the advantage of hindsight, we can see that his method does lead to the definition. Many standard results of elementary matrix theory first appeared long before matrices were the object of mathematical investigation. For example de Witt in Elements of curves , published as a part of the commentaries on the 1660 Latin version of Descartes ' Géométrie , showed how a transformation of the axes reduces a given equation for a conic to canonical form. This amounts to diagonalising a symmetric matrix but de Witt never thought in these terms. The idea of a determinant appeared in Japan and Europe at almost exactly the same time although Seki in Japan certainly published first. In 1683 Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables in exactly the
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way the Chinese methods described above were constructed. Without having any word which corresponds to 'determinant' Seki still introduced determinants and gave general methods for calculating them based on examples. Using his 'determinants' Seki was able to find determinants of 2 2, 3 3, 4 4 and 5 5 matrices and applied them to solving equations but not systems of linear equations. Rather remarkably the first appearance of a
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This note was uploaded on 03/30/2008 for the course MAT 2240 taught by Professor Greenwald during the Spring '08 term at Appalachian State.

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Matrices and Determinants - Matrices and Determinants...

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