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

Question

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 .

Answer 1

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

Answer 2

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 $n \geq 2$.

Answer 3

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 $n\geq 2$.

Answer 4

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 $2$.