Characteristic equation of square matrix of order 4 and 5 in terms of determinants of its sub matrices

The method to find the characteristic equation of a square matrix of any order is to solve the equation det. (A – ) = 0 i.e. | A – | = 0 which takes so much time when n = 4,5,…..& so on. This paper is an attempt to simplify the method to find the characteristic equation of a square matrix A of order 4 and 5 by taking the determinants of its all submatrices starting from 1x1 upto nxn.

Introduction

Definitions Square matrix: A rectangular matrix [aij]mxn of order m x n is said to be square matrix if m = n Determinant: Determinant of a square matrix is a number associated to it. Characteristic equation: If A=[aij]nxn is square matrix of order n then equation |A-ߣI| = 0 is its characteristic equation

Result

If A= [aij]nxn is square matrix of order n then its characteristic equation is given by |A-λI| = 0 i.e. (-1)nλn+ (-1)n-1∑ |a୧୧| ୬ ୧ୀଵ λn-1 + (-1)n-2∑ ∑ ሺi ൏jሻฬ a୧୧ a୧୨ a୨୧ a୨୨ฬ ୬ ୨ୀଶ ୬ିଵ ୧ୀଵ λn-2 + (-1)n-3∑ ∑ ∑ ሺi ൏j ൏kሻอ a୧୧ a୧୨ a୧୩ a୨୧ a୨୨ a୨୩ a୩୧ a୩୨ a୩୩ อ ୬ ୩ୀଷ ୬ିଵ ୨ୀଶ ୬ିଶ ୧ୀଵ λn-3 + (-1)n-4λn-4∑ ∑ ∑ ∑ ሺi ൏j ൏k ൏lሻተ ተ ୬ ୪ୀସ ୬ିଵ ୩ୀଷ ୬ିଶ ୨ୀଶ ୬ିଷ ୧ୀଵ +... +(-1)0λ0|A| = 0 a୧୧ a୧୨ a୧୩ a୧୪ a୨୧ a୨୨ a୨୩ a୨୪ a୩୧ a୩୨ a୩୩ a୩୪ a୪୧ a୪୨ a୪୩ a୪୪ Corllary 1: For A = ൦ ൪, aଵଵ aଵଶ aଵଷ aଵସ aଶଵ aଶଶ aଶଷ aଶସ aଷଵ aଷଶ aଷଷ aଷସ aସଵ aସଶ aସଷ aସସ the characteristic equation is (-1)4λ4+ (-1)4-1∑ |a୧୧| ସ ୧ୀଵ λ4-1 + (-1)4-2∑ ∑ ሺi ൏jሻฬ a୧୧ a୧୨ a୨୧ a୨୨ฬ ସ ୨ୀଶ ସିଵ ୧ୀଵ λ4-2 + (-1)4-3∑ ∑ ∑ ሺi ൏ ସ ୩ୀଷ ସିଵ ୨ୀଶ ସିଶ ୧ୀଵ j ൏kሻอ a୧୧ a୧୨ a୧୩ a୨୧ a୨୨ a୨୩ a୩୧ a୩୨ a୩୩ อλ4-3 + (-1)0λ0|A| = 0 Or (-1)4 λସ + (-1)3 (a11+ a22+ a33+ a44)λଷ. Maths 2017; 2(6): 99-102 Ashwani Jain Assistant Professor in Mathematics, Sgg Janta Girls College, Raikot, Punjab, India Correspondence Ashwani Jain Assistant Professor in Mathematics, Sgg Janta Girls College, Raikot, Punjab, India +(-1)2(ቚaଵଵ aଵଶ aଶଵ aଶଶቚ+ቚaଵଵ aଵଷ aଷଵ aଷଷቚ+ቚaଵଵ aଵସ aସଵ aସସቚ+ቚaଶଶ aଶଷ aଷଶ aଷଷቚ+ቚaଶଶ aଶସ aସଶ aସସቚ+ቚaଷଷ aଷସ aସଷ aସସቚ)λଶ + (-1)1 (อ aଵଵ aଵଶ aଵଷ aଶଵ aଶଶ aଶଷ aଷଵ aଷଶ aଷଷ อ+อ aଵଵ aଵଶ aଵସ aଶଵ aଶଶ aଶସ aସଵ aସଶ aସସ อ+อ aଵଵ aଵଷ aଵସ aଷଵ aଷଷ aଷସ aସଵ aସଷ aସସ อ+อ aଶଶ aଶଷ aଶସ aଷଶ aଷଷ aଷସ aସଶ aସଷ aସସ อ)λଵ + (-1)0ተ ተλ଴= 0 aଵଵ aଵଶ aଵଷ aଵସ aଶଵ aଶଶ aଶଷ aଶସ aଷଵ aଷଶ aଷଷ aଷସ aସଵ aସଶ aସଷ aସସ Example: Take A = ൦ ൪ 3 െ3 5 1 1 െ2 3 0 4 5 2 9 െ2 െ5 െ1 1 Then characteristic equation of A is given by |A – λ I | = 0 or ተ ተ = 0 3 െλ െ3 5 1 1 െ2 െλ 3 0 4 5 2 െλ 9 െ2 െ5 െ1 1 െλ or ሺ3 െλሻ อ െ2 െλ 3 0 5 2 െλ 9 െ5 െ1 1 െλ อ + 3อ 1 3 0 4 2 െλ 9 െ2 െ1 1 െλ อ + 5 อ 1 െ2 െλ 0 4 5 9 െ2 െ5 1 െλ อ - 1อ 1 െ2 െλ 3 4 5 2 െλ െ2 െ5 െ1 อ = 0 or (3- λ)(- λଷ + λଶ + 10 λ - 172) + 3 (λଶ + 9 λ – 55) + 5(- 4λଶ+ 9λ + 94) -1(- 2λଶ- 9λ -25) = 0 or λସ - 4λଷ - 22λଶ + 283 λ - 186 = 0 Using the above result (-1)4 λସ + (-1)3 (3+(-2)+2+1)λଷ +(-1)2(ቚ3 െ3 1 െ2ቚ൅ቚ3 5 4 2ቚ൅ቚ3 1 െ2 1ቚ൅ቚെ2 3 5 2ቚ൅ቚെ2 0 െ5 1ቚ൅ቚ2 9 െ1 1ቚ)λଶ + (-1)1 (อ 3 െ3 5 1 െ2 3 4 5 2 อ+อ 3 െ3 1 1 െ2 0 െ2 െ5 1 อ+อ 3 5 1 4 2 9 െ2 െ1 1 อ+อ െ2 3 0 5 2 9 െ5 െ1 1 อ)λଵ ተ= 0 + ተ 3 െ3 5 1 1 െ2 3 0 4 5 2 9 െ2 െ5 െ1 1 We get the same result λସ - 4λଷ - 22λଶ + 283 λ - 186 = 0 , Corllary 2: For A = ۑ ۑ ۑ ې ۏ ێ ێ ێ ۍ aଵଵ aଵଶ aଵଷ aଵସ aଵହ aଶଵ aଶଶ aଶଷ aଶସ aଶହ aଷଵ aଷଶ aଷଷ aଷସ aଷହ aସଵ aସଶ aସଷ aସସ aସହ aହଵ aହଶ aହଷ aହସ aହହے the characteristic equation is (-1)5λ5+ (-1)5-1∑ |a୧୧| ହ ୧ୀଵ λ5-1 + (-1)5-2∑ ∑ ሺi ൏jሻฬ a୧୧ a୧୨ a୨୧ a୨୨ฬ ହ ୨ୀଶ ହିଵ ୧ୀଵ λ5-2 + (-1)5-3∑ ∑ ∑ ሺi ൏j ൏kሻอ a୧୧ a୧୨ a୧୩ a୨୧ a୨୨ a୨୩ a୩୧ a୩୨ a୩୩ อ ହ ୩ୀଷ ହିଵ ୨ୀଶ ହିଶ ୧ୀଵ λ5-3 +(-1)5-4∑ ∑ ∑ ∑ ሺi ൏j ൏k ൏lሻተ ተ ହ ୪ୀସ ହିଵ ୩ୀଷ ହିଶ ୨ୀଶ ହିଷ ୧ୀଵ λ5-4+(-1)5-5λ5-5|A| = 0 a୧୧ a୧୨ a୧୩ a୧୪ a୨୧ a୨୨ a୨୩ a୨୪ a୩୧ a୩୨ a୩୩ a୩୪ a୪୧ a୪୨ a୪୩ a୪୪ Or (-1)5 λହ + (-1)4 (a11+ a22+ a33+ a44+ a55)λସ +(-1)3(ቚaଵଵ aଵଶ aଶଵ aଶଶቚ+ቚaଵଵ aଵଷ aଷଵ aଷଷቚ+ቚaଵଵ aଵସ aସଵ aସସቚ+ቚaଵଵ aଵହ aହଵ aହହቚ+ቚaଶଶ aଶଷ aଷଶ aଷଷቚ+ቚaଶଶ aଶସ aସଶ aସସቚ+ ቚaଶଶ aଶହ aହଶ aହହቚ+ቚaଷଷ aଷସ aସଷ aସସቚ+ቚaଷଷ aଷହ aହଷ aହହቚ+ቚaସସ aସହ aହସ aହହቚ)λଷ + (-1)2 (อ aଵଵ aଵଶ aଵଷ aଶଵ aଶଶ aଶଷ aଷଵ aଷଶ aଷଷ อ+อ aଵଵ aଵଶ aଵସ aଶଵ aଶଶ aଶସ aସଵ aସଶ aସସ อ+อ aଵଵ aଵଶ aଵହ aଶଵ aଶଶ aଶହ aହଵ aହଶ aହହ อ+อ aଵଵ aଵଷ aଵସ aଷଵ aଷଷ aଷସ aସଵ aସଷ aସସ อ+อ aଵଵ aଵଷ aଵହ aଷଵ aଷଷ aଷହ aହଵ aହଷ aହହ อ +อ aଵଵ aଵସ aଵହ aସଵ aସସ aସହ aହଵ aହସ aହହ อ+อ aଶଶ aଶଷ aଶସ aଷଶ aଷଷ aଷସ aସଶ aସଷ aସସ อ+อ aଶଶ aଶଷ aଶହ aଷଶ aଷଷ aଷହ aହଶ aହଷ aହହ อ+อ aଶଶ aଶସ aଶହ aସଶ aସସ aସହ aହଶ aହସ aହହ อ+อ aଷଷ aଷସ aଷହ aସଷ aସସ aସହ aହଷ aହସ aହହ อ)λଶ + (-1)1(ተ ተ+ተ ተ+ተ ተ aଵଵ aଵଶ aଵଷ aଵହ aଶଵ aଶଶ aଶଷ aଶହ aଷଵ aଷଶ aଷଷ aଷହ aହଵ aହଶ aହଷ aହହ aଵଵ aଵଶ aଵସ aଵହ aଶଵ aଶଶ aଶସ aଶହ aସଵ aସଶ aସସ aସହ aହଵ aହଶ aହସ aହହ aଵଵ aଵଶ aଵଷ aଵସ aଶଵ aଶଶ aଶଷ aଶସ aଷଵ aଷଶ aଷଷ aଷସ aସଵ aସଶ aସଷ aସସ ተ+ተ +ተ ተ)λଵ+(-1)0ተተ ተተλ଴= 0 aଵଵ aଵଷ aଵସ aଵହ aଷଵ aଷଷ aଷସ aଷହ aସଵ aସଷ aସସ aସହ aହଵ aହଷ aହସ aହହ aଶଶ aଶଷ aଶସ aଶହ aଷଶ aଷଷ aଷସ aଷହ aସଶ aସଷ aସସ aସହ aହଶ aହଷ aହସ aହହ aଵଵ aଵଶ aଵଷ aଵସ aଵହ aଶଵ aଶଶ aଶଷ aଶସ aଶହ aଷଵ aଷଶ aଷଷ aଷସ aଷହ aସଵ aସଶ aସଷ aସସ aସହ aହଵ aହଶ aହଷ aହସ aହହ Example: Take A = ۑ ۑ ۑ ې ۏ ێ ێ ێ ۍെ2 3 5 6 െ1 9 7 െ1 0 3 െ4 െ2 1 3 2 5 9 3 2 3 4 5 6 1 2 ے Then characteristic equation of A is given by |A – λ I | = 0 Or ተተ ተተ = 0 െ2 െλ 3 5 6 െ1 9 7 െλ െ1 0 3 െ4 െ2 1 െλ 3 2 5 9 3 2 െλ 3 4 5 6 1 2 െλ Or (-2- λ) ተ ተ-3ተ ተ +5ተ ተ-6ተ ተ 9 െ1 0 3 െ4 1 െλ 3 2 5 3 2 െλ 3 4 6 1 2 െλ 7 െλ െ1 0 3 െ2 1 െλ 3 2 9 3 2 െλ 3 5 6 1 2 െλ 9 7 െλ 0 3 െ4 െ2 3 2 5 9 2 െλ 3 4 5 1 2 െλ 9 7 െλ െ1 3 െ4 െ2 1 െλ 2 5 9 3 3 4 5 6 2 െλ +(-1) ተ ተ = 0 9 7 െλ െ1 0 െ4 െ2 1 െλ 3 5 9 3 2 െλ 4 5 6 1 Or (-2- λ)(λସ-12λଷ+2λଶ+192λଵ-288)-3(-9λଷ+29λଶ+276λଵ-112)+5(-4λଷ+49λଶ+164λଵ+16)-6(-5λଷ-31λଶ+84λଵ+720)-1(4λଷ-14λଶ+ 160λଵ+608) = 0 Or -λହ + 10λସ+55λଷ+162λଶ-768λଵ-3936 = 0 Use the above result (-1)5 λହ + (-1)4 (-2+7+1+2+2)λସ +(-1)3(ቚെ2 3 9 7ቚ+ቚെ2 5 െ4 1ቚ+ቚെ2 6 5 2ቚ+ቚെ2 െ1 4 2 ቚ+ቚ7 െ1 െ2 1 ቚ+ቚ7 0 9 2ቚ+ ቚ7 3 5 2ቚ+ቚ1 3 3 2ቚ+ቚ1 2 6 2ቚ+ቚ2 3 1 2ቚ)λଷ + (-1)2 (อ െ2 3 5 9 7 െ1 െ4 െ2 1 อ+อ െ2 3 6 9 7 0 5 9 2 อ+อ െ2 3 െ1 9 7 3 4 5 2 อ+อ െ2 5 6 െ4 1 3 5 3 2 อ+อ െ2 5 െ1 െ4 1 2 4 6 2 อ +อ െ2 6 െ1 5 2 3 4 1 2 อ+อ 7 െ1 0 െ2 1 3 9 3 2 อ+อ 7 െ1 3 െ2 1 2 5 6 2 อ+อ 7 0 3 9 2 3 5 1 2 อ+อ 1 3 2 3 2 3 6 1 2 อ)λଶ + (-1)1(ተ ተ+ተ ተ+ተ ተ +ተ ተ+ተ ተ)λଵ+(-1)0ተተ ተተλ଴= 0 െ2 3 6 െ1 9 7 0 3 5 9 2 3 4 5 1 2 7 െ1 0 3 െ2 1 3 2 9 3 2 3 5 6 1 2 െ2 3 5 6 9 7 െ1 0 െ4 െ2 1 3 5 9 3 2 െ2 3 5 െ1 9 7 െ1 3 െ4 െ2 1 2 4 5 6 2 െ2 5 6 െ1 െ4 1 3 2 5 3 2 3 4 6 1 2 െ2 3 5 6 െ1 9 7 െ1 0 3 െ4 െ2 1 3 2 5 9 3 2 3 4 5 6 1 2 Or (-1)5 λହ + (-1)4 (10)λସ+(-1)3(-55)λଷ+(-1)2(162)λଶ+(-1)1(768)λଵ+(-1)0(-3936)λ଴= 0 Or -λହ + 10λସ+55λଷ+162λଶ-768λଵ-3936 = 0

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