Block symmetric matrix expressed with Kronecker product and its determinant My ultimate goal is to find a way to compute the determinant of the following block symmetric matrix:
$$
    \underset{np\times np}H=\begin{bmatrix}
    \frac{1}{n}A+\frac{2(n-1)}{n^3s^2}I_p & -\frac{2}{n^3s^2}I_p & \dots & -\frac{2}{n^3s^2}I_p \\
    -\frac{2}{n^3s^2}I_p & \frac{1}{n}A+\frac{2(n-1)}{n^3s^2}I_p & \dots & -\frac{2}{n^3s^2}I_p \\
\dots & \dots & \dots & \dots\\
    -\frac{2}{n^3s^2}I_p & -\frac{2}{n^3s^2}I_p & \cdots & \frac{1}{n}A+\frac{2(n-1)}{n^3s^2}I_p\\
    \end{bmatrix},
$$
where $A$ is a $p\times p$ symmetric matrix and $n,s\in \mathbb{R}$. 
As a first step, I would like to express this matrix with a unique expression envolving Kronecker product. Any idea of how to achieve such expression? Or any suggestion to compute the determinant of $H$ in another way?
 A: We can write
$$
H = \frac 1n  I_n \otimes A + M \otimes I_p
$$
where the entries of $M$ are the coefficients to your $I_p$.  That is,
$$
M = \pmatrix{
\frac{2(n-1)}{n^3s^2} & -\frac{2}{n^3s^2} & \cdots & -\frac{2}{n^3s^2} \\
    -\frac{2}{n^3s^2} & \frac{2(n-1)}{n^3s^2} & \cdots & -\frac{2}{n^3s^2} \\
\vdots & \vdots & \ddots & \vdots\\
    -\frac{2}{n^3s^2} & -\frac{2}{n^3s^2} & \cdots & \frac{2(n-1)}{n^3s^2}\\
}
$$
You could say that your $H$ is the Kronecker sum $M \oplus \frac{A}{n}$. 
One approach to finding the determinant is as follows: you can find the eigenvalues of $M$ with one of the methods outlined here.  If $\lambda_1,\dots,\lambda_n$ are the eigenvalues of $M$ and $\mu_1,\dots,\mu_p$ are the eigenvalues of $\frac An$, then the eigenvalues of $H$ will be $\lambda_i + \mu_j$ for every pair $i,j$ with $1 \leq i \leq n$ and $1 \leq j \leq p$.  So, the determinant of $H$ is the product of the eigenvalues of $H$, that is the product of all such sums.

Here is a reference for the Kronecker sum. See section 2.1.3 in particular.
