Hello All
Consider a matrix with elements:
$A_{i,j}=x_i$ for $i=j$
$A_{i,j}=1$ for $i\neq j$
Is there a closed form expression for the elements of $A^{1}$?
Will be glad to know of any reference.
Thanks
HC
Hello All Consider a matrix with elements: $A_{i,j}=x_i$ for $i=j$ Is there a closed form expression for the elements of $A^{1}$? Will be glad to know of any reference. Thanks HC 


You can use the ShermanMorrison formula. In the notation of the Wikipedia article, let $u=v=(1,\ldots,1)'$ and $A$ (not the same as your $A$) be the diagonal matrix with $(x_{1}1, \ldots, x_{n}1)$ on the diagonal. Then, if I haven't made a mistake, the entry of the inverse matrix you're looking for is $\frac{1}{x_{i}1}  \frac{ \frac{1}{(x_{i}1)^{2}} }{ 1 + \sum_{k} \frac{1}{x_{k}1}}$ if $i=j$, and $ \frac{ \frac{1}{ (x_{i}1)(x_{j}1) }}{ 1 + \sum_{k} \frac{1}{x_{k}1}}$ otherwise. 

