• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.609-615
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• 2012

In the current paper, we establish the relationship between the powers of an invertible matrix and the powers of its inverse. More precisely, we prove that if A is an invertible matrix and, if $A^n\;=\;(A_{i,j}(n))$ for all positive integer n, then $A^{-n}\;=\;(A_{i,j}{(-n))$.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.1-7
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• 2006

In this paper, we investigate bounded matrices over regular rings. We observe that every bounded matrix over a regular ring can be described by idempotent matrices and invertible matrices. Let A, $B{in}M_n(R)$ be bounded matrices over a regular ring R. We prove that $(AB)^d = U(BA)^dU^{-1}$ for some $U{\in}GL_n(R)$.

• Journal of the Korean Mathematical Society
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• v.46 no.2
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• pp.257-269
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• 2009

A ring R is a weakly stable ring provided that aR + bR = R implies that there exists $y\;{\in}\;R$ such that $a\;+\;by\;{\in}\;R$ is right or left invertible. In this article, we characterize weakly stable rings by virtue of $2{\times}2$ invertible matrices over them. It is shown that a ring R is a weakly stable ring if and only if for any $A\;{\in}GL_2(R)$, there exist two invertible lower triangular L and K and an invertible upper triangular U such that A = LUK, where two of L, U and K have diagonal entries 1. Related results are also given. These extend the work of Nagarajan et al.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.14 no.6
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• pp.105-110
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• 2004

MAC is used for data origin authentication or message integrity protection. In Crypto'03 Cary and Venkatesan introduced new MAC based on unimodular matrix groups. It is to encrypt messages using private keys and to encrypt them again using public keys which are matrices whose determinants are $\pm$1. These matrices have property called k-invertible. This k effects on the collision probability of this new MAC. The smaller k is, the less collisions occur. Cary shows 6-invertible matrices, and 10-invertible matrices whose components are only 1, 0, -1. In this paper we figure out sufficient conditions about choosing 4 matrices among special 22 matrices. Also, we introduce 5-invertible matrices whose components are 1, 0, -1. Those have better efficiency and security.

• East Asian mathematical journal
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• v.23 no.2
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• pp.207-212
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• 2007

We describe the class of invertible matrices T such that $TAT^{-1}$ is EPr, for a given EPr matrix A of order n. Necessary and sufficient condition is determined for $TAT^{-1}$ to be EP for an arbitrary matrix A of order n.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.873-884
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• 2009

We characterize QB-ideals of exchange rings by means of quasi-invertible elements and annihilators. Further, we prove that every $2\times2$ matrix over such ideals of a regular ring admits a diagonal reduction by quasi-inverse matrices. Prime exchange QB-rings are studied as well.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.577-581
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• 2002

The incidence matrices corresponding to a nil-algebra of finite index % can be used to determine the nilpotency. We find the smallest positive integer n such that the sum of the incidence matrices Σ$\_$p/$\^$p/ is invertible. In this paper, we give a different proof of the case that the nil-algebra of index 2 has nilpotency less than or equal to 4.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.643-652
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• 2008

We construct the sets of Boolean matrix pairs, which are naturally occurred at the extreme cases for the Boolean rank inequalities relative to the sums and difference of two Boolean matrices or compared between their Boolean ranks and their real ranks. For these sets, we consider the linear operators that preserve them. We characterize those linear operators as T(X) = PXQ or $T(X)\;=\;PX^tQ$ with appropriate invertible Boolean matrices P and Q.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1139-1150
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• 2009

We obtain necessary and sufficient conditions for $2{\tims}2$ block operator valued matrices to be Moore-Penrose (MP) invertible and give new representations of such MP inverses in terms of the individual blocks.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.387-396
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• 2016

An $n{\times}n$ matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let $A(x){\in}M_n(R[[x]])$. We prove, in this note, that $A(x){\in}M_n(R[[x]])$ is strongly clean if and only if $A(0){\in}M_n(R)$ is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.