Assume the matrix of order n is :$\displaystyle A = \left(\begin{array}{ccc}a_{11}&...&a_{1n}\\...&... &...\\a_{n1}&...&a_{nn}\end{array}\right)$

and we have:$\displaystyle \sum_{j=1}^{n}a_{ij} = \sum_{j=1}^{n}a_{ji} = 0,i = 1,2,...,n.$

show that: all the algebraic complement(the number of algebraic complements is $\displaystyle n^{2}$ ) of$\displaystyle A$ are equal.