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Found problems: 1

1940 Putnam, B6
Tags: matrix , matrix determinant , linear algebra
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.