Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$

Hint: $A = \begin{bmatrix}1\\1\\1\\\vdots\\1\end{bmatrix} \begin{bmatrix}1&2&3&\dots&n\end{bmatrix} + \begin{bmatrix}1\\2\\3\\\vdots\\n\end{bmatrix} \begin{bmatrix}1 & 1 & 1 & \dots & 1\end{bmatrix}$

The next column is obtained from the previous one by adding the vector $e=\begin{pmatrix}1\\1\\ \vdots \\1\end{pmatrix}$, thus the span of the column vectors is generated by the first column and $e$. Hence the rank is $2$ if $n \geq 2$.

Subtract the first column from all the other ones to get $$\begin{pmatrix} 2& \cdots & j-1& \cdots & n-1\\ 3& \cdots & j-1&\cdots & n-1\\ \vdots& \vdots & \vdots &\vdots & \vdots \\ n+1& \cdots & j-1&\cdots & n-1 \end{pmatrix}$$

The rank of this matrix is clearly $2$, provided $n\geq 2$.