1988 Putnam B5

For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$
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{aligned}

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{aligned}

$$

对于正整数 $n$，令 $M_n$ 为 $2n+1$ 除以 $2n+1$

斜对称矩阵，其中第一个 $n$ 中的每个条目

主对角线下方的次对角线为 1，其余各

entries below the main diagonal is -1.找出有证据的等级

$M_n$。 (根据一种定义，矩阵的秩是

最大的 $k$ 使得存在一个非零的 $k \times k$ 子矩阵

行列式。)

人们可能会注意到

$$

\开始{对齐}

M_1 &= \left( \begin{array}{ccc} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0

\end{数组}\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).

\结束{对齐}

$$