Putnam 1940 B6

Prove that the determinant of the matrix $$\begin{pmatrix} a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n \ldots & \ldots & \ldots & \ldots & \ldots a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k \end{pmatrix}$$ is divisible by $k^{n-1}$ and find its other factor.