元素d_(ij)=(i+j-1)~2的n阶行列式的性质

The n-order Determinants with Entries d_(ij)=(i+j-1)~2

设n阶行列式D_n的元素d_(ij)=(i+j-1)~2,本文讨论了D_n的一些性质,概括为:D_n的任一二阶子式为负数;D_n的任一三阶子式为负偶数;D_n的任一四阶子式和四阶以上的子式均为零,特别D_n=0.还讨论了四个连续正整数的平方数之间的关系,并推出了计算任一三阶子式的简便公式.

In this paper,we study a class of n x n determinants D n,whose(i,j)th entry is(i+j-1)^2.A list of properties were verified which includes:any minor determinant of order 2 is negative;any minor determinant of order 3 is negative even;any minor determinant of order≥4 is 0,in particular,D n=0.Furthermore,we look into the relationship of any four consecutive perfect squares,and obtained a convenient computational formula for the calculation of all order-3 minor determinants.