Inverse of Combinatorial Matrix

Theorem
Let $$C_n$$ be the combinatorial matrix of order $n$ given by:


 * $$C_n = \begin{bmatrix}

x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end{bmatrix}$$

Then its inverse $$C_n^{-1} = \left[{b}\right]_n$$ can be specified as:
 * $$b_{ij} = \frac {-y + \delta_{ij}\left({x + ny}\right)} {x \left({x + ny}\right)}$$

where $$\delta_{ij}$$ is the Kronecker delta.

Proof
We have from the definition that:
 * $$C_n = x \mathbf{I}_n + y \mathbf{J}_n$$

where:
 * $$\mathbf{I}_n$$ is the Identity Matrix of order $n$;
 * $$\mathbf{J}_n$$ is the square ones matrix of order $n$.

From Square of Ones Matrix we have $$\mathbf{J}_n^2 = n \mathbf{J}_n$$.

Hence:

$$ $$ $$ $$ $$ $$ $$

So we have found a specification for the matrix which, when multiplied by $$C_n$$, yields $$\mathbf{I}_n$$.

By using the identities $$\mathbf{I}_n = \left[{\delta_{ij}}\right]_n$$ and $$\mathbf{J}_n = \left[{1}\right]_n$$ we obtain the stated result:
 * $$b_{ij} = \frac {-y + \delta_{ij}\left({x + ny}\right)} {x \left({x + ny}\right)}$$