A necessary condition and a sufficient condition for a matrix to be embeddable in an invertible matrix over a commutative ring are given. Furthermore, it is proved that a necessary and sufficient condition for a matrix of order n to be embeddable in an invertible matrix of order n+1 over a principal ideal domain is that the elements in the adjoint matrix of this matrix are relatively prime. Partial results obtained in this paper generalize the corresponding results for the ring of integers.
郭小芳, 谭宜家. 关于交换环上矩阵嵌入可逆矩阵的一些条件 [J]. 兴发pt登录学报, 2020, 37(11): 91-93. GUO Xiaofang, TAN Yijia. Some Conditions for a Matrix to be Embeddable in an Invertible Matrix over a Commutative Ring . Journal of Jilin Institute of Chemical Technology, 2020, 37(11): 91-93.