Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$ Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$.
My work : I noticed that A is symmetric . Hence all of its eigen vectors are real .. That is all i have got . 
Your help will be highly appreciated .Thank you .
 A: Hint: $A = \begin{bmatrix}1\\1\\1\\\vdots\\1\end{bmatrix} \begin{bmatrix}1&2&3&\dots&n\end{bmatrix} + \begin{bmatrix}1\\2\\3\\\vdots\\n\end{bmatrix} \begin{bmatrix}1 & 1 & 1 & \dots & 1\end{bmatrix}$
A: 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 $n \geq 2$.
A: 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 $n\geq 2$.
A: If $n=1$, $r(A) = 1$. 
Otherwise if $n>1$, then $r(A) = 2$. Notice that 
$$A = 
\begin{pmatrix}
2 & 3 & \cdots & n+1 \\
3 & 4 & \cdots & n+2 \\
&&\cdots&\\
n+1&n+2&\cdots &2n 
\end{pmatrix}
$$
Use elementary row operations to subtract the 1st row from the $i$th row for $2 \leq i \leq n$. Then we get
$$ A' = \begin{pmatrix}
2 & 3 & \cdots & n+1 \\
1 & 1 & \cdots & 1 \\
2 & 2 & \cdots & 2 \\
&& \cdots & \\
n-1 & n-1 & \cdots & n-1 
\end{pmatrix} $$
And this matrix has rank $2$. Since we only used elementary row operations, $A$ also has rank $2$. 
