On a Method to Compute the Determinant of a 4 4 Matrix

Transcription

1 Interntionl Journl of Scientific nd Innovtive Mthemticl Reserch (IJSIMR) Volume 5 Issue PP 1-5 ISSN X (Print) & ISSN (Online) DOI: wwwrcjournlsorg On Method to Compute the Determinnt of 4 4 Mtrix Rez Frhdin frhdinrez@yhoocom Abstrct: In this rticle we will study n interesting method to compute the determinnt of squre mtrix of order 4 Keywords: Determinnt Mtrices of order 4 Duplex frction Dodgson s condenstion 1 INTRODUCTION In liner lgebr nd mtrix theory the determinnt of squre mtrix is very importnt for mny sciences s Physics Sttistics Engineering etc The determinnt of n n n mtrix n A [ n n1 n2 nn n n is denoted by det(a) or A nd bsic formul to compute the determinnt is n det(a) n sgn(j 1 j 2 j n ) 1j1 njn n1 n2 nn where the summtion is tken over ll n! permuttions j 1 j 2 j n of the set of integers 1 2 n Also the function sgn(j 1 j 2 j n ) is defined s: sgn(j 1 j 2 j n ) { +1 if j 1 j 2 j n is n even permuttion 1 if j 1 j 2 j n is n odd permuttion There re mny methods nd rules to compute the determinnt of squre mtrix nd some wellknown methods re Srrus rule tringle s rule Chio s condenstion method Dodgson s condenstion method etc In 2016 R Frhdin [2 estblished new method to compute the determinnt of squre mtrix of order 4 In this rticle we will study the Frhdin's method we lso present some new results For the next section the following nottion will be used: () A n denotes squre mtrix of order n (b) R denotes the set of ll rel numbers Interntionl Journl of Scientific nd Innovtive Mthemticl Reserch (IJSIMR)Pge 1

2 Rez Frhdin 2 DUPLEX FRACTION METHOD TO COMPUTE THE DETERMINANT OF A SQUARE MATRIX OF ORDER 4 In 2016 R Frhdin estblished the following definition (see [2 Definition 21): Definition 1 ( Duplex Frction ) Let nd b 11 b 12 re two 2 2 determinnts such 22 b 22 tht b 11 b 12 b 21 b 22 re nonzero numbers nd b 11 b 12 b 21 b 22 0 The duplex frction of on b 11 b 12 is defined s: b 21 b 22 b b11 b b 11 b b21 b22 12 b 11 b 12 b 21 b 22 b 21 b For exmple the duplex frction of on is Now we shll know bout the Dodgson s condenstion of mtrix tht ws introduced by Chrles Lutwidge Dodgson ( ) in 1866 (see [1): Definition 2 (Dodgson s condenstion) The Dodgson s condenstion of n n n mtrix A n [ ij n n is n (n 1) (n 1) mtrix such s [m ij (n 1) (n 1) such tht ij i(j+1) m ij (i+1)j (i+1)(j+1) Henceforth the nottion DC(A n ) denotes the Dodgson s condenstion of mtrix A n nd the second condenstion is DC(DC(A n )) nd so on Thus for 4 4 mtrix the firs Dodgson s condenstion is DC(A 4 ) nd the second condenstion is A 4 [ [ (1) Interntionl Journl of Scientific nd Innovtive Mthemticl Reserch (IJSIMR)Pge 2

3 On Method to Compute the Determinnt of 4 4 Mtrix DC(DC(A 4 )) [ (2) For exmple let using (1) we hve nd using (2) we hve A 4 [ DC(A 4 ) [ DC(DC(A 4 )) [ We hve the following theorem to compute the determinnt of squre mtrix of order 4 Theorem 1 Given 4 4 mtrix A 4 [ where re nonzero numbers nd Then Proof See [2 A 4 DC(DC(A 4 )) As n exmple of Theorem 1 consider the mtrix we hve A 4 [ DC(A 4 ) [ DC(DC(A 4 )) [ Interntionl Journl of Scientific nd Innovtive Mthemticl Reserch (IJSIMR)Pge 3

4 Rez Frhdin hence the determinnt of A 4 is equl to A We shll now prove the following theorem: Theorem 2 Given 4 4 mtrix A 4 [ 1 b where b 1 b 2 b 3 b 4 re nonzero numbers nd b 1 b 2 0 Then A 4 ( )(b 1 b 4 b 2 b 3 ) Proof We hve Using Theorem 1 we hve DC(DC(A 4 )) [ ( 1b 1 )(b 1 b 4 b 2 b 3 ) ( 2 b 2 )(b 1 b 4 b 2 b 3 ) ( 3 b 3 )(b 1 b 4 b 2 b 3 ) ( 4 b 4 )(b 1 b 4 b 2 b 3 ) 2 2 DC(DC(A 4 )) A 4 b 1 b 2 ( 1b 1 )(b 1 b 4 b 2 b 3 ) ( 2 b 2 )(b 1 b 4 b 2 b 3 ) ( 3 b 3 )(b 1 b 4 b 2 b 3 ) ( 4 b 4 )(b 1 b 4 b 2 b 3 ) b 1 b 2 ( 1 b 1 )(b 1 b 4 b 2 b 3 ) ( 2 b 2 )(b 1 b 4 b 2 b 3 ) b 1 b 2 ( 3 b 3 )(b 1 b 4 b 2 b 3 ) ( 4 b 4 )(b 1 b 4 b 2 b 3 ) b 1 b 2 1(b 1 b 4 b 2 b 3 ) 2 (b 1 b 4 b 2 b 3 ) 3 (b 1 b 4 b 2 b 3 ) 4 (b 1 b 4 b 2 b 3 ) b 1 b (b 1 b 4 b 2 b 3 ) (b 1 b 4 b 2 b 3 ) 2 b 1 b 4 b 2 b 3 ( )(b 1 b 4 b 2 b 3 ) Theorem 3 We hve 1 x y b 2 0 x b 1 b b y 0 3 b x y b 2 x 0 y ( )(b 1 b 4 b 2 b 3 ) where b 1 b 2 b 3 b 4 re nonzero numbers nd b 1 b 2 0 nd x y R Proof The proof is similr to the proof of Theorem 2 Interntionl Journl of Scientific nd Innovtive Mthemticl Reserch (IJSIMR)Pge 4

5 On Method to Compute the Determinnt of 4 4 Mtrix REFERENCES [1 C L Dodgson Condenstion of Determinnts Being New nd Brief Method for Computing their Arithmetic Vlues Proc Roy Soc Ser A 15 (1866) [2 R Frhdin Duplex Frction Method to Compute the Determinnt of 4 4 Mtrix 2016 vilble t Cittion: R Frhdin "On Method to Compute the Determinnt of 4 4 Mtrix" Interntionl Journl of Scientific nd Innovtive Mthemticl Reserch vol 5 no 4 p / Copyright: 2017 Authors This is n open-ccess rticle distributed under the terms of the Cretive Commons Attribution License which permits unrestricted use distribution nd reproduction in ny medium provided the originl uthor nd source re credited Interntionl Journl of Scientific nd Innovtive Mthemticl Reserch (IJSIMR)Pge 5