I have the following task: Let $K$ be a field, $n \in \mathbb{N}$ and $a,b \in K^n$. Evaluate the determinant of the following matrix:
$$\begin{pmatrix} a_1+b_1 & b_2 & b_3 & \dots & b_n& \\ b_1 & a_2 + b_2 & b_3 & \dots & b_n \\ b_1 & b_2 & a_3 + b_3 & \dots & b_n \\ \vdots & \vdots & \vdots& & \vdots \\ b_1 & b_2 & b_3 &\dots & a_n + b_n \end{pmatrix}$$
What I did was expanding it as follows using the Laplace expansion:
$$\det A =(a_1 + b_1) \det\begin{pmatrix} a_2 + b_2 &\dots& b_n \\ b_2 &\dots& b_n \\ \vdots & & \vdots \\ b_2 & \dots & a_n + b_n \end{pmatrix} - b_2 \det\begin{pmatrix} b_1 & b_3 &\dots& b_n \\ b_1 & a_3 + b_3 &\dots& b_n \\ \vdots & \vdots & & \vdots \\ b_1 & b_3 & \dots & a_n + b_n \end{pmatrix} + \ b_3 \det\begin{pmatrix} \dots \end{pmatrix} - \dots (-1)^{n+1}b_n \det\begin{pmatrix} b_1 &\dots& b_{n-1} \\ b_1 &\dots& b_{n-1} \\ \vdots & & \vdots \\ b_1 & \dots & b_{n-1} \end{pmatrix}$$
And before I expand the rest of those determinants and fill 20 papers with a's and b's I'd like to ask for advice. Is this the right way? And when I think about it I don't really see any simplification that is possible when I have finally expanded everything to a point where I could use Cramers rule. It just came to my mind that I could also expand using the Lapace rule by iterating through the rows instead of the columns. By doing that I'd be able to factor out all of those $b_1$...
NOTE: I am not allowed to use the Sylverster Determinant Theorem
Thank you very much for your help.
FunkyPeanut