MathDB

For positive integers

n

, let

M_n

be the

2n+1

2n+1

skew-symmetric matrix for which each entry in the first

n

subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of

M_n

. (According to one definition, the rank of a matrix is the largest

k

such that there is a

k \times k

submatrix with nonzero determinant.)
One may note that \begin{align*} M_1 &= \left( \begin{array}{ccc} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{array}\right) \\ M_2 &= \left( \begin{array}{ccccc} 0 & -1 & -1 & 1 & 1 \\ 1 & 0 & -1 & -1 & 1 \\ 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 1 & 0 & -1 \\ -1 & -1 & 1 & 1 & 0 \end{array} \right). \end{align*}

Problem Statement