This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

中文翻译：

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对偶矩阵什么时候有对偶广义逆？

本文讨论了对偶矩阵的各种类型的对偶广义逆的存在。获得了关于存在各种类型对偶广义逆的充要条件的新的和基础的结果。结果表明，与实矩阵不同，对偶矩阵可能没有{1}-对偶广义逆矩阵。得到了对偶矩阵具有{1}-对偶广义逆的充要条件。结果表明，对偶矩阵总是具有 {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-对偶广义逆如果并且仅当它具有 {1}-对偶广义逆并且每个对偶矩阵都具有 {2}- 和 {2,4}-对偶广义逆时。提供了所有这些对偶逆的明确表达，迄今为止文献中尚未报道。证明了对偶矩阵的 Moore-Penrose 对偶广义逆存在当且仅当对偶矩阵具有 {1}-对偶广义逆；这个对偶逆的显式表达式，当它存在时，不管它的实部的等级如何。还获得了对偶矩阵的 Moore-Penrose 对偶逆的显式表达式，根据乘积的 {1}-对偶广义逆。还提供了一些与使用限制较少的对偶逆元确定对偶 Moore-Penrose 逆元相关的新结果。还获得了{1}-对积的对偶广义逆。还提供了一些与使用限制较少的对偶逆元确定对偶 Moore-Penrose 逆元相关的新结果。还获得了{1}-对积的对偶广义逆。还提供了一些与使用限制较少的对偶逆元确定对偶 Moore-Penrose 逆元相关的新结果。