On the Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with Fibonacci and Lucas Numbers

Transcription

1 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog O the Determiats ad Iverses of Skew Circulat ad Skew Left Circulat Matrices with Fiboacci ad Lucas Numbers YUN GAO Liyi Uiversity Departmet of Mathematics Shuaglig Road Liyi CHINA gaoyu2014@126com ZHAOLIN JIANG Liyi Uiversity Departmet of Mathematics Shuaglig Road Liyi CHINA jzh1208@siacom YANPENG GONG Liyi Uiversity Departmet of Mathematics Shuaglig Road Liyi CHINA gyp2011@siacom Abstract: I this paper we cosider the skew circulat ad skew left circulat matrices with the Fiboacci ad Lucas umbers Firstly we discuss the ivertibility of the skew circulat matrix ad preset the determiat ad the iverse matrix by costructig the trasformatio matrices Furthermore the ivertibility of the skew left circulat matrices are also discussed We obtai the determiats ad the iverse matrices of the skew left circulat matrices by utilizig the relatio betwee skew left circulat matrices ad skew circulat matrix respectively Key Words: Skew circulat matrix Skew left circulat matrix Determiat Iverse Fiboacci umber Lucas umber 1 Itroductio Skew circulat ad circulat matrices have importat applicatios i various disciplies icludig image processig commuicatios sigal processig ecodig solvig Toeplitz matrix problems precoditioer ad solvig least squares problems They have bee put o firm basis with the work of P Davis [1] ad Z L Jiag [2] The skew-circulat matrices as pre-coditioers for liear multistep formulaelmf-based ordiary differetial equatiosodes codes Hermitia ad skew-hermitia Toeplitz systems are cosidered i [ ] Lyess employed a skew-circulat matrix to costruct s-dimesioal lattice rules i [7] Spectral decompositios of skew circulat ad skew left circulat matrices are discussed i [8] The Fiboacci ad Lucas sequeces are defied by the followig recurrece relatios respectively: F +1 F + F 1 F 0 0 F 1 1 L +1 L + L 1 L 0 2 L 1 1 for 0: The first few values of the sequeces are give by the followig table: F L The {F } is give by the formula F α β α β ad the {L } is give by the formula L α + β α ad β are the roots of the characteristic equatio x 2 x 1 0 Besides some scholars have give various algorithms for the determiats ad iverses of osigular circulat matrices [1] Ufortuately the computatioal complexity of these algorithms are very amazig with the order of matrix icreasig However Some authors gave the explicit determiats ad iverse of circulat ad skew-circulat ivolvig Fiboacci ad Lucas umbers For example D V Jaiswal evaluated some determiats of circulat whose elemets are the geeralized Fiboacci umbers [9] D A Lid preseted the determiats of circulat ad skew-circulat ivolvig Fiboacci umbers [10] D Z Li gave the determiat of the Fiboacci-Lucas quasi-cyclic matrices i [11] S Q She cosidered circulat matrices with Fiboacci ad Lucas umbers ad preseted their explicit determiats ad iverses by costructig the trasformatio matrices [12] The purpose of this paper is to obtai the better results for the determiats ad iverses of skew circulat ad skew left circulat matrices by some perfect properties of Fiboacci ad Lucas umbers I this paper we adopt the followig two covetios ad for ay sequece {a } ki a k 0 i the case i > Defiitio 1 [8] A skew circulat matrix with the first E-ISSN: Issue 4 Volume 12 April 2013

2 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog row a 1 a 2 a is meat a square matrix of the form a 1 a 2 a a a 1 a 1 a 1 a a 2 a 2 a 3 a 1 deoted by SCirca 1 a 2 a Defiitio 2 [8] A skew left circulat matrix with the first row a 1 a 2 a is meat a square matrix of the form a 1 a 2 a a 2 a 3 a 1 a 3 a 4 a 2 a a 1 a 1 deoted by SLCirca 1 a 2 a Lemma 3 [8 1] Let A SCirca 1 a 2 a be skew circulat matrix the we have i A is ivertible if ad oly if fω k η 0k fx j1 a jx j1 ω exp 2πi πi ad η exp ; ii If A is ivertible the the iverse of A is a skew circulat matrix Lemma 4 With the orthogoal skew left circulat matrix Θ : it holds that SCirca 1 a 2 a ΘSLCirca 1 a 2 a Lemma 5 If the [SCirca 1 a 2 a ] 1 SCircb 1 b 2 b [SLCirca 1 a 2 a ] 1 SLCircb 1 b b 2 Proof: Let B SCirca 1 a 2 a A SLCirca 1 a 2 a by Lemma 4 we have B ΘA the B 1 A 1 Θ 1 Thus we obtai A 1 B 1 Θ SLCircb 1 b b 2 B 1 SCircb 1 b 2 b 2 Determiat ad iverse of skew circulat matrix with the Fiboacci umbers I this sectio let A SCircF 1 F 2 F be skew circulat matrix Firstly we give a determiat explicit formula for the matrix A Afterwards we prove that A is a ivertible matrix for 2 ad the we fid the iverse of the matrix A Theorem 6 Let A SCircF 1 F 2 F be skew circulat matrix the we have det A 1 + F k1 + F 2 F+1 F k 1 F F is the th Fiboacci umber Proof: Obviously det A 1 1 satisfies the equatio 1 I the case > 1 let Γ F F 1 +F Π 1 0 F F 1 +F F 0 F 1 +F be two matrices the we have ΓA Π 1 F 1 f b 13 b 11 b 1 0 f b 23 b 21 b b F b F b E-ISSN: Issue 4 Volume 12 April 2013

3 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog b 1j F j+2 b 2j F j+1 b jj F 1 + F +1 j k+1 F f F 1 + F + F k F 1 + F +1 ad 1 k+1 f F F k+1 F 1 + F +1 So we obtai det Γ det A det Π 1 2 ] k+1 F F 1 [F 1 + F + F k F 1 + F +1 F 1 + F ] k+1 F F 1 [F 1 + F +1 + F k F 1 + F +1 F 1 + F F F k1 F+1 F k F while hece we have det Γ det Π det A 1 + F k1 F 2 F+1 F k F The proof is completed Theorem 7 Let A SCircF 1 F 2 F be skew circulat matrix if 2 the A is a ivertible matrix Proof: Whe 3 i Theorem 6 the we have det A hece A 3 is ivertible I the case > 3 sice F α β αβ α + β 1 αβ 1 ω exp 2πi fω k η πi ad η exp hece we have F j ω k η j1 j1 1 α β α j β j ω k η j1 j1 1 [ α1 + α α β 1 αω k η β1 + ] β 1 βω k η 1 [ α β + α +1 β +1 α β 1 α + βω k η + αβω 2k η 2 αβα β ω k ] η 1 α + βω k η + αβω 2k η F +1 + F ω k η 1 ω k η ω 2k η 2 k If there exists ω l ηl such that fω l η 0 the we obtai 1 + F +1 + F ω l η 0 for 1 ω l η ω 21 η 2 0 thus ω l η 1+F +1 F is a real umber While 2l + 1πi ω l η exp 2l + 1π 2l + 1π cos + i si hece si 2l+1π 0 so we have ω l η 1 for 0 < 2l+1π < 2π But x 1 is t the root of the equatio 1 + F +1 + F x 0 > 3 Hece we obtai fω k η 0 for ay ω k η k while fη j1 F jη j1 1+F +1+F η 0 1ηη 2 Hece by Lemma 3 the coclusio is obtaied Lemma 8 Let the matrix G [g ij ] 2 ij1 be of the form F 1 + F +1 i j g ij F i j otherwise The the iverse G 1 [g ij ]2 ij1 of the matrix G is equal to { g ij F ij F 1 +F +1 i j ij+1 0 i < j Proof: Let c ij 2 g ikg kj The c ij 0 for i < j I the case i j we obtai c ii g ii g ii F 1 + F +1 1 F 1 + F +1 1 E-ISSN: Issue 4 Volume 12 April 2013

4 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog For i j + 1 we obtai 2 c ij g ik g kj g ii1g i1j + g ii g ij F F ij1 F 1 + F +1 ij + F 1 + F +1 F ij 0 F 1 + F +1 ij+1 Hece we verify GG 1 I 2 I 2 is 2 2 idetity matrix Similarly we ca verify G 1 G I 2 Thus the proof is completed Theorem 9 Let A SCircF 1 F 2 F be skew circulat matrix if 2 the we have 2 x 1 1 A 1 1 f SCircx 1 x 2 x F i F i1 F 1 + F +1 i 2 x F 1i F i1 2 1 F 1 + F +1 i x k F k3 k 3 4 F 1 + F +1 k2 2 k+1 F f F 1 + F + F k F 1 + F +1 Proof: Let 1 f π 13 π 14 π 1 F F 3 F f f 1 f Π π 1j f f F j+1 F j+2 2 f F 1 + F + F k ad j 3 4 F F 1 + F +1 1 k+1 f F F k+1 F 1 + F +1 k+1 The we have ΓA Π 1 Π 2 D 1 G D 1 diagf 1 f is a diagoal matrix ad D 1 G is the direct sum of D 1 ad G If we deote Π Π 1 Π 2 the we obtai A 1 ΠD 1 1 G 1 Γ Sice the last row elemets of the matrix Π are 0 1 F 2 f F 3 f F 2 f F 1 f Hece by Lemma 8 if let A 1 SCircx 1 x 2 x the its last row elemets are give by the followig equatios: x F 1i F i1 f f F 1 + F +1 i x 3 1 F 1 f F 1 + F +1 x 4 1 f x 5 1 f 2 3 F 3i F i1 F 1 F 1 + F +1 i f F 1 + F +1 F 4i F i1 F 1 + F +1 i 1 f F 1 f F 1 + F +1 x 1 2 F 1i F i1 f F 1 + F +1 i 1 3 f F 2i F i1 F 1 + F +1 i 1 4 F 3i F i1 f F 1 + F +1 i 2 x F 1i F i1 f f F 1 + F +1 i 1 3 F 2i F i1 f F 1 + F +1 i F 3i F i1 F 1 + F +1 i Let C j j F j+1i F i1 F 1 +F +1 j i the we have C 2 C 1 F 1 + F 1 + F +1 F F 1 + F F 3i F i1 F 1 + F +1 i E-ISSN: Issue 4 Volume 12 April 2013

5 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog ad C C 3 F i F i1 3 F i2 F i1 F 1 + F +1 i + F 1 + F +1 i F 1F 3 F 1 + F C j+2 2 j+2 j+1 F i F i1 F 1 + F +1 i C j+1 C j F j+3i F i1 F 1 + F +1 i F j+2i F i1 F 1 + F +1 i F i F i1 F 1 + F +1 i j F 2 F j F 1 + F +1 j+1 + F 1F j+1 F 1 + F +1 j+2 F j+1i F i1 F 1 + F +1 i F 1F j F 1 + F +1 j+1 j F j+5i 2F j+4i + F j+2i F i1 + F 1 + F +1 i F j+1 j F 1 + F +1 j+2 Hece we obtai x 1 1 C2 f f C 3 x 2 C f x 3 C1 1 f f F i F i1 F 1 + F +1 i f x 4 C2 C 1 1 f f 1 1 F 1 + F +1 x 5 C3 C 2 C 1 1 f f x C2 1 f C 3 f F 3 F 1 + F F 1i F i1 F 1 + F +1 i F F 1 + F +1 2 F 2 C 4 F 1 + F +1 3 ad A 1 1 f SCircx 1 x 2 x 2 x 1 1 F i F i1 F 1 + F +1 i 2 x F 1i F i1 2 1 F 1 + F +1 i x k F k3 k 3 4 F 1 + F +1 k2 The proof is completed 3 Determiat ad iverse of skew circulat matrix with the Lucas umbers I this sectio let B SCircL 1 L 2 L be skew circulat matrix Firstly we give a determiat explicit formula for the matrix B Afterwards we prove that B is a ivertible matrix for ay positive iteger ad the we fid the iverse of the matrix B Theorem 10 Let B SCircL 1 L 2 L be skew circulat matrix the we have det B 1 + L L L k+1 L k L k L L is the the lucas umber Proof: Obviously det B 1 1 satisfies the equatio 2 Whe > 1 let Σ Ω L +2 L 1 +L L+2 L 1 +L L+2 L 1 +L E-ISSN: Issue 4 Volume 12 April 2013

6 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog be two matrices the we have ΣB Ω 1 L 1 l L 1 L 3 L 2 0 l c 23 c 21 c c L c L c ad c 2j 3L j+2 L j+3 c jj L 1 + L +1 j l L 1 + 3L + 3L k+1 L k+2 L + 2 k+1 L 1 + L +1 1 l L k+1 L + 2 k+1 L 1 + L +1 Hece we obtai det Σ det B det Ω 1 2 L 1 [L 1 + 3L + 3L k+1 L k+2 L + 2 k+1 ] L 1 + L +1 2 L 1 + L +1 1 L 1 [L 1 + L L k+1 L k+2 L + 2 k+1 ] L 1 + L L L L 2 2 while 3L k+1 L k+2 1 Thus we have 1 + L k L det Σ det Ω det B 1 + L L 2 2 3L k+1 L k+2 1 The proof is completed 1 + L k L Theorem 11 Let B SCircL 1 L 2 L be skew circulat matrix the B is ivertible for ay positive iteger Proof: Sice L α +β α+β 2 αβ 1 ω exp 2πi πi ad η exp hece we have fω k η L j ω k η j1 j1 α j + β j ω k η j1 j1 α1 + α 1 αω k η + β1 + β 1 βω k η α + β + α+1 + β +1 1 α + βω k η + αβω 2k η 2 αβα + β ω k η + 2αβω k η 1 α + βω k η + αβω 2k η L L ω k η 1 2ω k η ω 2k η 2 k If there exists ω l ηl such that fω l η 0 the we obtai 2+L L ω l η 0 for 1 2ω l η ω 2l η 2 0 thus ω l η 2+L +1 L +2 is a real umber While 2l + 1πi ω l η exp 2l + 1π cos + i si 2l + 1π Hece si 2l+1π 0 so we have ω l η 1 for 0 < 2l+1π < 2π But x 1 is t the root of the equatio 2 + L L x 0 for ay positive iteger Hece we obtai fω k η 0 for ay ω k ηk while fη j1 L jη j1 2+L +1+2+L η 0 Thus by 12ηη 2 Lemma 3 the coclusio is obtaied Lemma 12 Let the matrix H [h ij ] 2 form h ij ij1 L 1 + L +1 i j 2 + L i j otherwise the the iverse H 1 [h ij ]2 equal to h ij ij1 { L2 ij L 1 +L +1 ij+1 i j 0 i < j be of the of the matrix H is E-ISSN: Issue 4 Volume 12 April 2013

7 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog Proof: Let r ij 2 h ikh kj Obviously r ij 0 for i < j I the case i j we obtai r ii h ii h ii L 1 + L +1 For i j + 1 we obtai 1 L 1 + L r ij h ik h kj h ii1h i1j + h ii h ij 2 + L L 2 ij1 L 1 + L +1 ij + L 1 + L +1 L 2 ij 0 L 1 + L +1 ij+1 Hece HH 1 I 2 I 2 is 2 2 idetity matrix Similarly we ca verify H 1 H I 2 Thus the proof is completed Theorem 13 Let B SCircL 1 L 2 L be skew circulat matrix the we have 2 y 1 1 B 1 1 l SCircy 1 y 2 y L +2i 3L +1i L 2 i1 L 1 + L +1 i 2 y 2 L +1i 3L i L 2 i1 3 L 1 + L +1 i y k 5L 2 k3 k 3 4 L 1 + L +1 k2 Proof: Let 1 l ω 13 ω 14 ω ω 23 ω 24 ω Ω ω 1j l 3L j+2 L j+3 l L j+2 ω 2j L j+3 3L j+2 j 3 4 l 2 l L 1 + 3L + 3L k+1 L k+2 L + 2 k+1 L 1 + L +1 ad 1 l L k+1 L + 2 k+1 L 1 + L +1 The we have ΣB Ω 1 Ω 2 D 2 H D 2 diagl 1 l is a diagoal matrix ad D 2 H is the direct sum of D 2 ad H If we deote Ω Ω 1 Ω 2 the we obtai B 1 ΩD 1 2 H 1 Σ Sice the last row elemets of the matrix Ω are 0 1 L 3L 1 l L 13L 2 l L 33L 2 l Hece by Lemma 12 if let B 1 SCircy 1 y 2 y the its last row elemets are give by the followig equatios: y L +1i 3L i L 2 i1 l l L 1 + L +1 i y 3 1 L 3 3L 2 l L 1 + L +1 y 4 1 L 3 3L 2 l L 1 + L l L 5i 3L 4i L 2 i1 L 1 + L +1 i y 5 1 L 3 3L 2 l L 1 + L L 5i 3L 4i L 2 i1 l L 1 + L +1 i + 1 l 3 L 6i 3L 5i L 2 i1 L 1 + L +1 i y 1 4 L 1i 3L 2i L 2 i1 l L 1 + L +1 i 1 3 L i 3L 1i L 2 i1 l L 1 + L +1 i L +1i 3L i L 2 i1 l L 1 + L +1 i y L i 3L 1i L 2 i1 l l L 1 + L +1 i 1 2 L +1i 3L i L 2 i1 l L 1 + L +1 i E-ISSN: Issue 4 Volume 12 April 2013

8 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog Let D j j j the we have L j+3i 3L j+2i L 2 i1 L 1 + L +1 i D 1 D 2 L 3 3L 2 L 1 + L +1 2 L 5i 3L 4i L 2 i1 L 1 + L +1 i ad D j D L + 2 L 1 + L D 2 L i 3L i1 L 2 i1 L 1 + L +1 i L +1i 3L i L 2 i1 L 1 + L +1 i L 4 3L 3 L 2 3 L 1 + L D j+1 j j+1 L +2i 3L +1i L 2 i1 L 1 + L +1 i L +2i 3L +1i L 2 i1 L 1 + L +1 i D j+2 L j+3i 3L j+2i L 2 i1 L 1 + L +1 i L j+4i 3L j+3i L 2 i1 + L 1 + L +1 i j+2 L j+5i 3L j+4i L 2 i1 L 1 + L +1 i L 3 3L 2 L 2 j L 1 + L +1 j+1 L 4 3L 3 L 2 j L 1 + L +1 j+1 L 3 3L 2 L 2 j+1 L 1 + L +1 j+2 5L 2 j+1 j L 1 + L +1 j+2 Hece we obtai y 1 1 D3 1 l l 2 D 2 L +2i 3L +1i L 2 i1 1 L 1 + L +1 i y 2 D2 + 3 l 1 2 L +1i 3L i L 2 i1 3 l L 1 + L +1 i y 3 D1 1 5 l l L 1 + L +1 y 4 D1 D 2 1 5L 2 l l L 1 + L +1 2 y 5 D1 + D 2 D 3 1 5L 2 2 l l L 1 + L +1 3 y D4 ad + D 3 l 1 l 5L 2 3 L 1 + L y 1 1 D 2 B 1 1 l SCircy 1 y 2 y L +2i 3L +1i L 2 i1 L 1 + L +1 i 2 y 2 L +1i 3L i L 2 i1 3 L 1 + L +1 i y k 5L 2 k3 k 3 4 L 1 + L +1 k2 The proof is completed 4 Determiat ad iverse of skew left circulat matrix with the Fiboacci umbers I this sectio let A SLCircF 1 F 2 F be skew left circulat matrix By usig the obtaied coclusios i Sectio 2 we give a determiat explicit formula for the matrix A Afterwards we prove that E-ISSN: Issue 4 Volume 12 April 2013

9 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog A is a ivertible matrix for ay positive iteger The iverse of the matrix A is also preseted Accordig to Lemma 4 Lemma 5 Theorem 6 Theorem 7 ad Theorem 9 we ca obtai the followig theorems Theorem 14 Let A SLCircF 1 F 2 F be skew left circulat matrix the we have [ det A F F F+1 F k F 1 F is the th Fiboacci umber k1 ] Theorem 15 Let A SLCircF 1 F 2 F be skew left circulat matrix if > 2 the A is a ivertible matrix Theorem 16 Let A SLCircF 1 F 2 F > 2 be skew left circulat matrix the we have A 1 1 f SLCircx 1 x 2 x 2 k+1 F f F 1 + F + F k F 1 + F +1 2 x 1 1 x k F i F i1 F 1 + F +1 i F k1 k F 1 + F +1 k 2 x 1 + F 1i F i1 F 1 + F +1 i 5 Determiat ad iverse of skew left circulat matrix with Lucas umbers I this sectio let B SLCircL 1 L 2 L be skew left circulat matrix By usig the obtaied coclusios i Sectio 3 we give a determiat explicit formula for the matrix B Afterwards we prove that B is a ivertible matrix for ay positive iteger ad the we also fid the iverse of the matrix B Accordig to Lemma 4 Lemma 5 Theorem 10 Theorem 11 ad Theorem 13 we ca obtai the followig results Theorem 17 Let B SLCircL 1 L 2 L be skew left circulat matrix the we have [ det B L L 2 2 3L k+1 L k+2 1 L is the th Lucas umber 1 + L k1 ] L Theorem 18 Let B SLCircL 1 L 2 L be skew left circulat matrix the B is ivertible for ay positive iteger Theorem 19 Let B SLCircL 1 L 2 L be skew left circulat matrix the we have B 1 1 l SLCircy 1 y 2 y 2 l L 1 + 3L + 3L k+1 L k+2 L + 2 k+1 L 1 + L +1 2 y 1 1 y L +2i 3L +1i L 2 i1 L 1 + L +1 i k 5L 2 k1 k L 1 + L +1 k 2 y Coclusios L +1i 3L i L 2 i1 L 1 + L +1 i Besides some scholars have give various algorithms for the determiats ad iverses of osigular skew circulat matrices [12] For example the most commoly implemeted algorithms for computig the determiat ad iverse of osigular skew circulat matrix A SCirca 1 a 2 a are give by the followig formulas: ad deta fω j 1 2 j1 A 1 SCircb 1 b 2 b b s 1 r1 fωr ω r 1 2 s1 s 1 2 fx a ix i1 ad ω exp 2πi E-ISSN: Issue 4 Volume 12 April 2013

10 WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog Ufortuately the computatioal complexity of these algorithms are very amazig with the order of matrix icreasig For a geeral osigular skew circulat matrices its determiats ad iverses are hard determied oly by its first row The purpose of this paper is to obtai the explicit determiats ad iverses of skew circulat matrices by some perfect properties of Fiboacci ad Lucas umbers Refereces: [1] P J Davis Circulat Matrices Joh Wiley & Sos New York 1979 [2] Z L Jiag ad Z X Zhou Circulat Matrices Chegdu Techology Uiversity Publishig Compay Chegdu 1999 [3] D Bertaccii ad M K Ng Skew-circulat precoditioers for systems of LMF-based ODE codes Numer Aal Appl LNCS 2001 pp [4] R Cha X Q Ji Circulat ad skewcirculat precoditioers for skew-hermitia type Toeplitz systems BIT pp [5] R Cha M K Ng Toeplitz precoditioers for Hermitia Toeplitz sys- tems Liear Algebra Appl pp [6] T Huclke Circulat ad skew-circulat matrices for solvig Toeplitz matrix problems SIAM J Matrix Aal Appl pp [7] J N Lyess ad T S ö revik Four-dimesioal lattice rules geerated by skew-circulat matrices Math Comput pp [8] H Karer ad J Scheid ad C W Ueberhuber Spectral decompositio of real circulat matrices Liear Algebra Appl pp [9] D V Jaiswal O determiats ivolvig geeralized Fiboacci umbers Fiboacci Quart pp [10] D A Lid A Fiboacci circulat Fiboacci Quart pp [11] D Z Li Fiboacci-Lucas quasi-cyclic matrices Fiboacci Quart pp [12] S Q She ad J M Ce ad Y Hao O the determiats ad iverses of circulat matrices with Fiboacci ad Lucas umbers Appl Math Comput pp E-ISSN: Issue 4 Volume 12 April 2013