Find determinant $\det M$, where $m_{ij}=a_ia_j$, and $m_{ii}=a^2_i+k$

Question

Let ${a_1,\dots,a_n}$ --- sequence and $k\ne 0$. Define matrix $M$ in following way: $m_{ij}=a_ia_j$ if $i\ne j$, and $m_{ii}=a^2_i+k$. Find $\det M$.

Answer 1

Hint: $M=aa^T+kI$. To find $\det(M)$, you may determine $M$'s eigenvalues, or apply the matrix determinant formula for rank-1 update to a matrix.