Consider a matrix where the diagonal entries are anything but the off-diagonal entries are all one. I was able to find a formula for the determinant of this matrix, but what are other known properties? Does this matrix have a name? In particular is there a formula for its inverse?
Such a matriix has the form $J +D,$ where $D$ is a diagonal matrix, and $J$ is a square matrix with all entries $1$. One small remark is that if $D$ has two of its diagonal entries equal to $\lambda$, then $\lambda$ is also an eigenvalue of $J+D$. This is because the $\lambda$-eigenspace of $D$ is at least two-dimensional and the $0$-eigenspace of $J$ has codimension one.