Abstract
In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.
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
is given by
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 nonsymmetric 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
Let us assume now that
This means that
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).

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