BY 4.0 license Open Access Published by De Gruyter Open Access November 8, 2020

Explicit determinantal formula for a class of banded matrices

Yerlan Amanbek, Zhibin Du, Yogi Erlangga, Carlos M. da Fonseca, Bakytzhan Kurmanbek and António Pereira
From the journal Open Mathematics

Abstract

In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.

MSC 2010: 15A18; 15B05

1 Introduction

It was proved recently in [1] that the determinant of the banded matrix (which is a particular case of a Hessenberg matrix), for any integer n 4 ,

(1.1) A n = 1 1 0 0 a b 1 1 1 0 a 1 1 0 0 1 0 1 0 0 1 1 1 n × n

is given by

(1.2) det A n = ( a 1 ) 2 if n 0 ( mod 4 ) , a 2 + b + 1 if n 1 ( mod 4 ) , a 2 + 2 a b if n 2 ( mod 4 ) , a 2 if n 3 ( mod 4 ) ,

for any a and b. The proof for this equality is based on several auxiliary results established for particular cases of the matrix (1.1). As a corollary, two conjectures proposed in [2] are proved. For a recent and different approach, the reader is also referred to [3]. In this work, our goal is to provide a proof for (1.2) in a different way than [1]. The explicit formula for the determinant of the non-symmetric matrices can be applied in efficient computations, since several algorithms have been proposed to improve the efficiency of the determinant computation [4,5].

2 Proof

This new proof is based on the elementary properties of the determinant. First note that when n = 4 , 5 , 6 , 7 , one can deduce (1.2) by simple computations or by utilizing a Computer Algebra System such as Maple, Mathematica, and SAGE. For the convenience of the reader, we present the matrices for these cases,

A 4 = 1 1 a b 1 1 1 a 1 1 1 1 0 1 1 1 , A 5 = 1 1 0 a b 1 1 1 0 a 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 , A 6 = 1 1 0 0 a b 1 1 1 0 0 a 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 , A 7 = 1 1 0 0 0 a b 1 1 1 0 0 0 a 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 .

Let us assume now that n 8 . We used the cofactor expansion of det A n along the first column and the subtraction of the first row from second and third rows:

det A n = 1 1 0 0 a b 0 0 1 0 a a b 0 0 1 0 a b 0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 1 n × n (cofactor expansion along the first column)

= 0 1 0 0 a a b 0 1 0 a b 1 1 0 0 0 1 0 1 1 0 0 1 1 1 ( n 1 ) × ( n 1 ) (cofactor expansion along the first column)

= 1 0 0 0 a a b 1 1 0 0 a b 1 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 ( n 2 ) × ( n 2 ) ( R 2 R 1 and R 3 R 1 )

= 1 0 0 0 a a b 0 1 0 0 0 a 0 1 1 1 0 a b a 0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 1 ( n 2 ) × ( n 2 ) (cofactor expansion along the first column)

= 1 0 0 0 0 a 1 1 1 0 a b a 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 1 ( n 3 ) × ( n 3 ) ( R 2 R 1 and R 3 R 1 )

= 1 0 0 0 0 a 0 1 1 0 a b 0 1 1 1 0 0 a 0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 1 ( n 3 ) × ( n 3 ) (cofactor expansion along the first column)

= det A n 4 .

This means that det A n has period 4 and the proof is complete.

Acknowledgement

Yerlan Amanbek wishes to acknowledge the research Grant No. AP08052762, from the Ministry of Education and Science of the Republic of Kazakhstan and the FDCRG (Grant No. 110119FD4502). Zhibin Du was supported by the National Natural Science Foundation of China (Grant No. 11701505).

  1. Conflict of interest: C. M. da Fonseca is an Editor of the Open Mathematics and was not involved in the review process of this article.

References

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Received: 2020-05-15
Revised: 2020-09-14
Accepted: 2020-09-25
Published Online: 2020-11-08

© 2020 Yerlan Amanbek et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.