New Method to Calculate Determinants of n n(n 3) Matrix, by Reducing Determinants to 2nd Order

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1 International Journal of Algebra, Vol 6, 202, no 9, New Method to Calculate Determinants of n n(n 3) Matrix, by Reducing Determinants to 2nd Order Armend Salihu Department of Telecommunication, Faculty of Electrical and Computer Engineering, University of Prishtina, Bregu i Diellit pn, 0000 Prishtina, Kosovo arsalihu@gmailcom Abstract In this paper we will present a new method to calculate of n n (n 3) order determinants This method is based on Dodgson - Chio s condensation method, but the priority of this method compared with Dodgson - Chio s and minors method as well is that those method decreases the order of determinants for one, and this new method automatically affects in reducing the order of determinants in 2nd order Mathematics Subject Classification: 65F40, C20, 5A5 Keywords: New method to calculate determinants of n n matrix Introduction Let A be an n n matrix a a 2 a n a 2 a 22 a 2n A a n a n2 a nn Definition A determinant of order n, or size n n, (see [2], [3], [7], [8]) is the sum a a 2 a n a 2 a 22 a 2n D det(a) A ε j,j 2,,j n a j a j2 a jn, S n a n a n2 a nn ranging over the symmetric permutation group S n, where

2 94 A Salihu ε j,j 2,,j n { +, if j,j 2,,j n is an even permutation, if j,j 2,,j n is an odd permuation Chio s condensation method Chio s condensation is a method for evaluating an n n determinant in terms of (n ) (n ) determinants; see [4], [5]: A a a 2 a n a 2 a 22 a 2n a n a n2 a nn a n 2 a a 2 a 2 a 22 a a 2 a 3 a 32 a a 2 a n a n2 a a 3 a 2 a 23 a a 3 a 3 a 33 a a 3 a n a n3 a a 2 a a 3 a a n a n a 2n a n a 3n a n a nn 2 Dodgson s condensation method Dodgson s condensation method computes determinants of size n n by expressing them in terms of those of size (n ) (n ), and then expresses the latter in terms of determinants of size (n 2) (n 2), and so on (see [6]) 2 A new method This method is based on Dodgson and Chio s method, but the diference between them is that this new method is resolved by calculating 4 unique determinants of (n ) (n ) Order, (which can be derived from determinants of n n order, if we remove first row and first column or first row and last column or last row and first column or last row and last column, elements that belongs to only one of unique determinants we should call them unique elements), and one determinant of (n 2) (n 2) order which is formed from n n order determinant with elements a i,j with i, j,n, on condition that the determinant of (n 2) (n 2) 0 Theorem : Every determinant of n n (n > 2) order can be reduced into 2 2 order determinant, by calculating 4 determinants of (n ) (n ) order, and one determinant of (n 2) (n 2) order, on condition that (n 2) (n 2) order determinants to be different from zero Ongoing is presented a scheme of calculating the determinants of n n order according to this formula:

3 Method to compute n n determinant 95 a a 2 a n a 2 a 22 a 2n A a n a n2 a nn B C D E F, B 0 The B is (n 2) (n 2) order determinant which is the interior determinant of determinant A while C, D, E and F are unique determinants of (n ) (n ) order, which can be formed from n n order determinant Proof: Lets be n 4, and we will prove that the same result we can achieve when we calculate this determinant according to the above scheme: A a a 2 a 3 a 4 a 2 a 3 a 4 a 22 a 23 a 23 a 33 a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 a 2 a 22 a 23 a 3 a 32 a 33 a 4 a 42 a 43 a 2 a 3 a 4 a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 a 22 a 33 a 23 a 22 a 2 a 22 a 23 a 3 a 32 a 33 a 4 a 42 a 43 a 22 a 33 a 23 a 22 (A A 2 ) A A 2 (a 22 a 33 a 23 a 32 )(a a 22 a 33 a 44 +a a 23 a 34 a 42 +a a 24 a 32 a 43 a a 22 a 34 a 43 a a 23 a 32 a 44 a a 24 a 33 a 42 +a 2 a 23 a 3 a 44 +a 3 a 2 a 32 a 44 a 3 a 22 a 3 a 44 a 2 a 2 a 33 a a 2 a 2 a 34 a 43 +a 3 a 3 a 24 a 42 a 4 a 2 a 32 a 43 a 2 a 23 a 34 a 4 a 3 a 24 a 32 a 4 a 4 a 22 a 33 a 4 + +a 2 a 24 a 33 a 4 +a 3 a 22 a 34 a 4 +a 4 a 23 a 32 a 4 a 4 a 23 a 3 a 42 +a 4 a 2 a 33 a 42 +a 4 a 22 a 3 a 43 a 2 a 24 a 3 a 43 a 3 a 2 a 34 a 42 ) A a 22 a 33 a 23 a 32 (A A 2 ) a 22 a 33 a 23 a 32 (a 22 a 33 a 23 a 32 )

4 96 A Salihu (a a 22 a 33 a 44 +a a 23 a 34 a 42 +a a 24 a 32 a 43 a a 22 a 34 a 43 a a 23 a 32 a 44 a a 24 a 33 a a 2 a 23 a 3 a 44 +a 3 a 2 a 32 a 44 a 3 a 22 a 3 a 44 a 2 a 2 a 33 a 44 +a 2 a 2 a 34 a 43 +a 3 a 3 a 24 a 42 a 4 a 2 a 32 a 43 a 2 a 23 a 34 a 4 a 3 a 24 a 32 a 4 a 4 a 22 a 33 a 4 +a 2 a 24 a 33 a 4 +a 3 a 22 a 34 a 4 + +a 4 a 23 a 32 a 4 a 4 a 23 a 3 a 42 +a 4 a 2 a 33 a 42 +a 4 a 22 a 3 a 43 a 2 a 24 a 3 a 43 a 3 a 2 a 34 a 42 ) a a 2 a 3 a 4 a 2 a 3 a 4 Based on this we can outcome to the result: all combinations from C D E F which does not contain one of ε j,j 2,,j n a j a j2 a jn combinations from B determinant, and does not contain one of unique elements, as a result of crossed multiplication, they should be eliminated between each other, while other combinations which contain one of ε j,j 2,,j n a j a j2 a jn combinations from B determinants, extract as common elements and after divided by determinant B we get the result of the given determinant Example: let be the 5 order determinant: A ( ) The same result we can achieve even by calculating this determinant in other methods

5 Method to compute n n determinant 97 References [] C L Dodgson, Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetic Values, Proc Roy Soc Ser A, 5(866), [2] D Hajrizaj, New method to compute determinant of a 3x3 matrix, International Journal of Algebra, Vol 3, 2009, no 5, 2-29 [3] E Hamiti, Matematika, Universiteti i Prishtines: Fakulteti Elektroteknik, Prishtine, (2000), [4] F Chió, Mémoire sur les fonctions connues sous le nom de résultantes ou de déterminants Turin: E Pons, 853 [5] H Eves, An Introduction to the History of Mathematics, pages 405 and 493, Saunders College Publishing, 990 [6] H Eves, Chio s Expansion, 36 in Elementary Matrix Theory, New York: Dover, (996), [7] http : //enwikipediaorg/wiki/dodgson condensation [8] Q Gjonbalaj, ASalihu, Computing the determinants by reducing the order by four, Applied Mathematics E-Notes, 0(200), 5-58 [9] R F Scott, The theory of determinants and their applications, Ithaca, New York: Cornell University Library, Cambridge: University Press, (904), 3-5 [0] S Barnard and J M Child, Higher Algebra, London Macmillan LTD New York, ST Martin s Press (959), 3 [] W L Ferrar, Algebra, A Text-Book of Determinants, Matrices, and Algebraic Forms, Second edition, Fellow and tutor of Hertford College Oxford, (957), 7 Received: April, 202