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1.) History

3rd century BC and 16thCentury

Determinants were first used in the Chinese mathematics textbook The Nine Chapters on the

Mathematical Art ,Chinese scholars, around the In Europe, 2 2 determinants were considered

by Cardanoat the end of the 16th century and larer ones by !eibni"#

17thCentury

In $apan, %eki &akaka"u is credited with the disco'ery with the resultant and determinant (at

first in 16)*, the complete 'ersion no later than 1+1-# In Europe, Cramer(1+.- added to the

theory, treatin the sub/ect in relation to sets of e0uations# &he recurrence law was first

announced by "out(1+63-#

It was 4andermonde(1++1- who first reconi"ed determinants as independent functions# !aplace

(1++2-521522a'e the eneral method of expandin a determinant in terms of its complementary

minors7 4andermonde had already i'en a special case# Immediately followin, !arane(1++*-

treated determinants of the second and third order# !arane was the first to apply determinants

to 0uestions of elimination theory8 he pro'ed many special cases of eneral identities#

18THCentury

9auss(1)1- made the next ad'ance# !ike !arane, he made much use of determinants in the

theory of numbers# :e introduced the word determinant(!aplace had used resultant-, thouh not

in the present sinification, but rather as applied to the discriminant of a 0uantic# 9auss also

arri'ed at the notion of reciprocal (in'erse- determinants, and came 'ery near the multiplication

theorem#

&he next contributor of importance is inet (1)11, 1)12-, who formally stated the theorem

relatin to the product of two matrices of mcolumns and nrows, which for the special case of m

; nreduces to the multiplication theorem#

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treats this sub/ect, as well as the class of alternatin functions which %yl'ester has called

alternants# >bout the time of $acobi@s last memoirs, %yl'ester (1)*B- and Cayleybean their

work#

&he study of special forms of determinants has been the natural result of the completion of the

eneral theory# >xisymmetric determinants ha'e been studied by !ebesue, :esse, and

%yl'ester8 persymmetric determinants by %yl'ester and :ankel8 circulants by Catalan,

%pottiswoode, 9laisher, and %cott8 skew determinants and faffians, in connection with the

theory of orthoonal transformation, by Cayley8 continuants by %yl'ester8 Aronskians(so called

by uir- byChristoffeland robenius8 compound determinants by %yl'ester, Feiss, and ic0uet8

$acobians and :essians by %yl'ester8 and symmetric auche determinants by &rudi# merica, :anus (1))6-, Aeld (1)B*-,

and uirGet"ler (1B**- published treatises#

2. Application of Matrices

Area of Trianle

!"a#ple$

ind the area of trianle whose 'ertices are (2, H+-, (1, *-, (1, )-#

%olution$

(x1, y1- ; (2, H+-,(x2, y2- ; (1, *-,(x*, y*- ; (1, )-

>rea of the trianle

; 12

; 12 52(* H )- +(1 H 1- 1() H *-

; 12 5H1 H 6* H 22

; 12 x (H B.- ; H 3+#.

%ince area has to be a positi'e 0uantity, it is i'en by 3+#. s0#units

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!"a#ple 2

%ol'e the system of linear e0uations#

x 2y *" ; 6

2x 3y " ; +

*x 2y B" ; 13

Cramer@s rule#

%olution$&he determinant of coefficients

D ; ; 2

; 25(1-(1) H 1- H (1-(1) H *- *(2 H 6-

; 251+ H 1. H 12 ; H2

D1;

c1J c1H (c2 c*-

; ; D ; H2

D2 ;

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c*; c*H ( c1 c2-

!"a#ple 3. Ho#oeneous !&uations 'Constant ( )

Consider the homoeneous e0uations

a1x b1y c1" ;

a2x b2y c2" ;

a*x b*y c*" ;

&he homoenous system of e0uations is always consistent because x ; , y ; , " ; satisfies all

the e0uations in the system# &his solution is called the tri'ial solution#

then the system has a nonHtri'ial solution also# In fact, it has an infinite number of solutions and

is said to be dependent#

then the system has only the tri'ial solution x ; , y ; , " ; #

!"a#ple *. +on Ho#oenous !&uations 'solution ,y the Matri" Method)

Consider the nonHhomoeneous e0uations

a1x b1y c1" ; d1

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a2x b2y c2" ; d2

a*x b*y c*" ; d*

&his can be written as

Fefferences7

http7GGwww#shelo'esmath#comGalebraGad'ancedHalebraGmatricesHandHsol'inHsystemsHwithH

matricesG

http7GGen#wikipedia#orGwikiGatrixK(mathematics-

http7GGmath#tutor'ista#comGalebraGapplicationHofHmatricesHandHdeterminants#html

http://www.shelovesmath.com/algebra/advanced-algebra/matrices-and-solving-systems-with-matrices/http://www.shelovesmath.com/algebra/advanced-algebra/matrices-and-solving-systems-with-matrices/http://en.wikipedia.org/wiki/Matrix_(mathematics)http://math.tutorvista.com/algebra/application-of-matrices-and-determinants.htmlhttp://www.shelovesmath.com/algebra/advanced-algebra/matrices-and-solving-systems-with-matrices/http://www.shelovesmath.com/algebra/advanced-algebra/matrices-and-solving-systems-with-matrices/http://en.wikipedia.org/wiki/Matrix_(mathematics)http://math.tutorvista.com/algebra/application-of-matrices-and-determinants.html