There are inequalities about rank of the sum of two matrices in advanced algebra. However, few rnpeople study the problems of inequality about rank of sum of some matrices and the conditions for equality sign in rnthe inequality. In this paper, we firstly discuss the inequalities about rank of sum of a number of matrices. Secondrnly, by mathematical induction and partition of the matrix we give the condition to hold the identity in the above inernquality, i.e. the rank of the sum of matrix equal to the sum of their rank if and only if they can be transformed synrnchronously into block diagonal matrix. Finally, we give its applications for two matrices in linear algebra.%高等代数中常见两个矩阵之和的秩不等式,但对于若干个矩阵的和之秩的不等式问题以及不等式中等号成立的条件讨论不多.首先利用向量组的线性相关性对不等式进行证明,然后利用数学归纳法以及矩阵的分块法证明不等式中等号成立的条件,即矩阵和的秩等于它们秩的和当且仅当它们可以同时变为块对角矩阵.最后给出两个矩阵时不等式等号成立的结论的一些具体应用.
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