The determinant $\left|\begin{array}{ccc}y+k& y& y\\ y& y+k& y\\ y& y& y+k\end{array}\right|$ is equal to:

$\begin{array}{l}\left|\begin{array}{ccl}y+k& y& y\\ y& y+k& y\\ y& y& y+k\end{array}\right|=(y+k)\left[(y+k{)}^2-y^2\right]-y\left[y(y+k)-y^2\right]+y\left[y^2-y(y+k)\right]\\ =(y+k{)}^3-y^2(y+k)-y^2(y+k)+y^3+y^3-y^2(y+k)\\ =y^3+k^3+3y^{2}k+3y{k}^2+2y^3-3y^3-3y^{2}k\\ =k^2(k+3y)\end{array}$