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Find the rank of the matrix:

$$\mathcal{M}=\begin{bmatrix} 1^2 &2^2 &3^2 &\cdots &n^2 \\ 2^2& 3^2 &4^2 &\cdots &\left ( n+1 \right )^2 \\ 3^2& 4^2 &5^2 &\cdots &\left ( n+2 \right )^2 \\ \vdots & \vdots &\vdots & \ddots &\vdots \\ n^2& \left ( n+1 \right )^2 & \left ( n+2 \right )^2 &\cdots &\left ( 2n-1 \right )^2 \end{bmatrix}$$

Adding and subtracting columns from other columns is obviously a procedure that does not affect the rank of the matrix. By first subtracting the \( n-1 \) -th column from the \( n \) -th, then the \( n-2 \) -th from the \( n-1 \) -th, and so on, and repeating the same procedure again, we get that \( \mathcal{M} \) has the same rank as the following matrix:
\begin{equation*}\mathcal{N} = \begin{bmatrix} 1^2 &2^2 &3^2 &\cdots &n^2 \\ 3& 5 &7 &\cdots &2n+1 \\ 2& 2 &2 &\cdots &2 \\ \vdots & \vdots &\vdots & \ddots &\vdots \\ 2& 2 & 2 &\cdots &2 \end{bmatrix}
\end{equation*}
which has rank 3.