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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

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system
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{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*}

Present the answer in LaTex format: \boxed{Your answer}
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