$\det(\mathbf{1} - c \mathbf{J}_n)$, where $\mathbf{J}_n$ is an $n\times n$ matrix of ones

Question

Let $\mathbf{J}_n$ be an $n\times n$ matrix of ones, and let $c$ be a real number. Compute the following determinant:

$$\det(\mathbf{1} - c \mathbf{J}_n)$$

where $\mathbf{1}$ denotes the $n\times n$ identity matrix.

$n$ is a positive integer, of course.

Answer 1

Hint: Show that $\det(id-cJ_n)=1-nc$ by induction. For $n=1$, $id-cJ_1=(1-c)$, so the determinant is $1-c$.

Another way is to see that $id-cJ_n$ is a symmetric matrix of a very special kind, where more general determinant formulas are known.