Good evening. There is a Matrix M $n\times n$, where $m_{ij} = a_{i}a_{j}$ when $i\neq j$ and $m_{ii} = a_{i}^2+k$. Find the determinant of the M.
I have found that for $n=1$: $$det(M_{1\times1})=k+a_1^2$$ for $n=2$: $$det(M_{2\times 2})=k^2+k(a_1^2+a_2^2)$$ for $n=3$: $$det(M_{3\times 3})=k^3+k^2(a_1^2+a_2^2+a_3^2)$$ As I understand for n=t will get $$det(M_{t\times t})=k^t+k^{t-1}(a_1^2+a_2^2+a_3^2+...a_{t}^2)$$
After that I tried to use mathematical induction, but it didn't help me. Thank you for help!