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

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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 p.n., 10000 Prishtina, Kosovo

ar.salihu@gmail.com Abstract

In this paper we will present a new method to calculate ofn×n(n≥ 3) order determinants. This method is based on Dodgson - Chio’s con-densation method, but the priority of this method compared with Dodg-son - 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, 11C20, 15A15 Keywords: New method to calculate determinants of n×n matrix

1 Introduction

Let A be an n×n matrix A = ⎡ ⎢ ⎢ ⎢ ⎣ a₁₁ a₁₂ · · · a₁_n a₂₁ a₂₂ · · · a₂_n .. . ... . .. ... a_n₁ a_n₂ · · · a_nn ⎤ ⎥ ⎥ ⎥ ⎦.

Deﬁnition 1. A determinant of order n, or size n×n, (see [2], [3], [7], [8]) is the sum D= det(A) =|A|= a₁₁ a₁₂ · · · a₁_n a₂₁ a₂₂ · · · a₂_n .. . ... . .. ... a_n₁ a_n₂ · · · a_nn = Sn ε_j₁_,j₂_{,... ,j}_na_j₁a_j₂. . . a_j_n,

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ε_j₁_,j₂_{,... ,j}_n =

+1, ifj₁, j₂, . . . , j_nis an even permutation −1, ifj₁, j₂, . . . , j_nis an odd permuation.

1.1 Chio’s condensation method

Chio’s condensation is a method for evaluating an n×n determinant in terms of (n−1)×(n−1) determinants; see [4], [5]: A = a₁₁ a₁₂ ... a₁_n a₂₁ a₂₂ ... a₂_n .. . ... . .. ... a_n₁ a_n₂ ... a_nn = 1 an₁₁−2 a₁₁ a₁₂ a₂₁ a₂₂ a₁₁ a₁₃ a₂₁ a₂₃ · · · a_a11 a1n 21 a2n a₁₁ a₁₂ a₃₁ a₃₂ a₁₁ a₁₃ a₃₁ a₃₃ · · · a_a11 a1n 31 a3n .. . ... . .. ... a₁₁ a₁₂ a_n₁ a_n₂ a₁₁ a₁₃ a_n₁ a_n₃ · · · a_a11 a1n n1 ann .

1.2 Dodgson’s condensation method

Dodgson’s condensation method computes determinants of size n×n by ex-pressing them in terms of those of size (n−1)×(n−1), 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 be-tween them is that this new method is resolved by calculating 4 unique deter-minants of (n−1)×(n−1) 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 ele-ments), 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 = 1, n, on condition that the determinant of (n−2)×(n−2)= 0.

Theorem 1: Every determinant ofn×n(n >2) order can be reduced into 2×2 order determinant, by calculating 4 determinants of (n−1)×(n−1) 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

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|A|= a₁₁ a₁₂ · · · a₁_n a₂₁ a₂₂ · · · a₂_n .. . ... . . . ... a_n₁ a_n₂ · · · a_nn = 1 |B| · |C| |D| |E| |F| ,|B| = 0.

The |B| is (n−2)×(n−2) order determinant which is the interior deter-minant of deterdeter-minant|A| while|C|, |D|,|E|and|F|are unique determinants of (n−1)×(n−1) 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|= a₁₁ a₁₂ a₁₃ a₁₄ a₂₁ a₂₂ a₂₃ a₂₄ a₃₁ a₃₂ a₃₃ a₃₄ a₄₁ a₄₂ a₄₃ a₄₄ = 1 a₂₂ a₂₃ a₂₃ a₃₃ · a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ a₁₂ a₁₃ a₁₄ a₂₂ a₂₃ a₂₄ a₃₂ a₃₃ a₃₄ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ a₄₁ a₄₂ a₄₃ a₂₂ a₂₃ a₂₄ a₃₂ a₃₃ a₃₄ a₄₂ a₄₃ a₄₄ = = 1 a₂₂a₃₃−a₂₃a₂₂· · ⎛ ⎝ a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ · a₂₂ a₂₃ a₂₄ a₃₂ a₃₃ a₃₄ a₄₂ a₄₃ a₄₄ − a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ a₄₁ a₄₂ a₄₃ · a₂₂ a₂₃ a₂₄ a₃₂ a₃₃ a₃₄ a₄₂ a₄₃ a₄₄ ⎞ ⎠= = 1 a₂₂a₃₃−a₂₃a₂₂ ·(A1−A2) A₁−A₂ = (a₂₂a₃₃−a₂₃a₃₂)·(a₁₁a₂₂a₃₃a₄₄+a₁₁a₂₃a₃₄a₄₂+a₁₁a₂₄a₃₂a₄₃−a₁₁a₂₂a₃₄a₄₃− −a₁₁a₂₃a₃₂a₄₄−a₁₁a₂₄a₃₃a₄₂+a₁₂a₂₃a₃₁a₄₄+a₁₃a₂₁a₃₂a₄₄−a₁₃a₂₂a₃₁a₄₄−a₁₂a₂₁a₃₃a₄₄+ +a₁₂a₂₁a₃₄a₄₃+a₁₃a₃₁a₂₄a₄₂−a₁₄a₂₁a₃₂a₄₃−a₁₂a₂₃a₃₄a₄₁−a₁₃a₂₄a₃₂a₄₁−a₁₄a₂₂a₃₃a₄₁+ +a₁₂a₂₄a₃₃a₄₁+a₁₃a₂₂a₃₄a₄₁+a₁₄a₂₃a₃₂a₄₁−a₁₄a₂₃a₃₁a₄₂+a₁₄a₂₁a₃₃a₄₂+a₁₄a₂₂a₃₁a₄₃− −a₁₂a₂₄a₃₁a₄₃−a₁₃a₂₁a₃₄a₄₂) |A|= 1 a₂₂a₃₃−a₂₃a₃₂ ·(A1−A2) = = 1 a₂₂a₃₃−a₂₃a₃₂ ·(a22a33−a23a32)·

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·(a₁₁a₂₂a₃₃a₄₄+a₁₁a₂₃a₃₄a₄₂+a₁₁a₂₄a₃₂a₄₃−a₁₁a₂₂a₃₄a₄₃−a₁₁a₂₃a₃₂a₄₄−a₁₁a₂₄a₃₃a₄₂+ +a₁₂a₂₃a₃₁a₄₄+a₁₃a₂₁a₃₂a₄₄−a₁₃a₂₂a₃₁a₄₄−a₁₂a₂₁a₃₃a₄₄+a₁₂a₂₁a₃₄a₄₃+a₁₃a₃₁a₂₄a₄₂− −a₁₄a₂₁a₃₂a₄₃−a₁₂a₂₃a₃₄a₄₁−a₁₃a₂₄a₃₂a₄₁−a₁₄a₂₂a₃₃a₄₁+a₁₂a₂₄a₃₃a₄₁+a₁₃a₂₂a₃₄a₄₁+ +a₁₄a₂₃a₃₂a₄₁−a₁₄a₂₃a₃₁a₄₂+a₁₄a₂₁a₃₃a₄₂+a₁₄a₂₂a₃₁a₄₃−a₁₂a₂₄a₃₁a₄₃−a₁₃a₂₁a₃₄a₄₂) = = a₁₁ a₁₂ a₁₃ a₁₄ a₂₁ a₂₂ a₂₃ a₂₄ a₃₁ a₃₂ a₃₃ a₃₄ a₄₁ a₄₂ a₄₃ a₄₄

Based on this we can outcome to the result: all combinations from |C| |D| |E| |F|

which does not contain one ofε_j₁_,j₂_{,... ,j}_na_j₁a_j₂. . . a_j_n combinations from|B| de-terminant, and does not contain one of unique elements, as a result of crossed multiplication, they should be eliminated between each other, while other com-binations which contain one of ε_j₁_,j₂_{,... ,j}_na_j₁a_j₂. . . a_j_n 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|= 1 0 3 5 1 0 1 5 1 0 0 4 0 0 2 2 3 1 2 0 1 0 0 1 1 = 1 1 5 1 4 0 0 3 1 2 · 1 0 3 5 0 1 5 1 0 4 0 0 2 3 1 2 0 3 5 1 1 5 1 0 4 0 0 2 3 1 2 0 0 1 5 1 0 4 0 0 2 3 1 2 1 0 0 1 1 5 1 0 4 0 0 2 3 1 2 0 0 0 1 1 =· · ·= = 1 −36· 140 170 −4 −64 = 1 −36 ·(−8960 + 680) = −8280 −36 = 230

The same result we can achieve even by calculating this determinant in other methods.

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References

[1] C. L. Dodgson, Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetic Values, Proc. Roy. Soc. Ser. A, 15(1866), 150-155.

[2] D. Hajrizaj, New method to compute determinant of a 3x3 matrix, Inter-national Journal of Algebra, Vol. 3, 2009, no. 5, 211 - 219

[3] E. Hamiti, Matematika 1, Universiteti i Prishtines: Fakulteti Elek-troteknik, Prishtine, (2000), 163-164.

[4] F. Chió, Mémoire sur les fonctions connues sous le nom de résultantes ou de déterminants. Turin: E. Pons, 1853.

[5] H. Eves, An Introduction to the History of Mathematics, pages 405 and 493, Saunders College Publishing, 1990.

[6] H. Eves, Chio’s Expansion, 3.6 in Elementary Matrix Theory, New York: Dover, (1996), 129-136.

[7] http://en.wikipedia.org/wiki/Dodgson condensation

[8] Q. Gjonbalaj, A.Salihu, Computing the determinants by reducing the order by four, Applied Mathematics E-Notes, 10(2010), 151 - 158

[9] R. F. Scott, The theory of determinants and their applications, Ithaca, New York: Cornell University Library, Cambridge: University Press, (1904), 3-5.

[10] S. Barnard and J. M. Child, Higher Algebra, London Macmillan LTD New York, ST Martin’s Press (1959), 131.

[11] W. L. Ferrar, Algebra, A Text-Book of Determinants, Matrices, and Alge-braic Forms, Second edition, Fellow and tutor of Hertford College Oxford, (1957), 7.