Finding dimension of a vector space Let $H_n$ be the space of all $n\times n$ matrices $A = (a_{i,j})$ with entries in $\mathbb{R}$ satisfying $a_{i,j} = a_{r,s}$ whenever $i+j = r+s$ $(i, j , r , s = 1, 2, \ldots, n)$. What would be dimension of $H_n$ as a vector space over $\mathbb{R}$?
i have options for the dimension
1 -  $n^2$
2-   $n^2-n+1$
3 - $2n+1$
4- $2n-1$
I am finding difficulty in identifying the matrix $A$ .
thanks for support
 A: The "lines" with $i+j$ constant can be visualized in the matrix as lines of slope $1$ (since we have our "$y$-axis" upside down):
$$\begin{array}{ccccc}
* & \circledast & + & \oplus &\cdots\\
\circledast & + & \oplus & \times & \cdots\\
+ & \oplus & \times & \otimes & \cdots\\
\oplus & \times & \otimes & \#& \cdots\\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{array}$$
All entries with $*$ have the same value, because their indices add up to $2$. All entries marked with $\circledast$ have the same value, because the indices add up to $3$. All entries marked with $+$ have the same value, because the indices add up to $4$. All entries marked with $\oplus$ are equal; all entries marked with $\times$ are equal; all entries marked with $\otimes$ are equal. Etc.
The possible values of $i+j$ range from $2$ (when $i=1=j$) all the way to $2n$ (when $i=n=j$). 
A: HINT: Pick $n$ of moderate size and write out an example, say
$$A=\pmatrix{a_{11}&a_{12}&a_{13}&a_{14}\\
a_{21}&a_{22}&a_{23}&a_{24}\\
a_{31}&a_{32}&a_{33}&a_{34}\\
a_{41}&a_{42}&a_{43}&a_{44}}\;.$$
What are the possible values of $i+j$ for an entry $a_{ij}$? Clearly $i+j$ ranges over the set $\{2,3,\dots,8\}$. For what sets of entries is $i+j$ constant?
$$A=\pmatrix{a_{11}&\color{red}{a_{12}}&\color{blue}{a_{13}}&\color{green}{a_{14}}\\
\color{red}{a_{21}}&\color{blue}{a_{22}}&\color{green}{a_{23}}&\color{purple}{a_{24}}\\
\color{blue}{a_{31}}&\color{green}{a_{32}}&\color{purple}{a_{33}}&\underline{a_{34}}\\
\color{green}{a_{41}}&\color{purple}{a_{42}}&\underline{a_{43}}&\bf a_{44}}\;.$$
Now generalize to arbitrary $n$.
