Evaluate determinant of an $n \times n$-Matrix 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 
 A: Note that your matrix can be written as
$$\text{diag}(a_1,a_2,\ldots,a_n) + \begin{bmatrix} 1\\1\\1\\ \vdots\\1\end{bmatrix} \begin{bmatrix} b_1 & b_2 & \cdots & b_n\end{bmatrix}$$
This is a rank $1$ update to a diagonal matrix, whose determinant can be computed using the Sylvester determinant theorem:
$$\det(I+UV^T) = \det(I+V^TU)$$
I will leave the rest to you since the question is tagged as a homework problem.
A: In the case $\Bbb K = \Bbb R$, you can use$$M_n(t)=\begin{vmatrix} 
a_1+tb_1 & tb_2 & tb_3 & \dots & tb_n& \\
tb_1 & a_2 + tb_2 & tb_3 & \dots & tb_n \\ 
tb_1 & tb_2 & a_3 + tb_3 & \dots & tb_n \\
\vdots & \vdots & \vdots& & \vdots \\
tb_1  & tb_2 & tb_3 &\dots & a_n + tb_n 
\end{vmatrix}$$
$$\begin{array}{ll}M_n'(t)&=\begin{vmatrix} 
b_1 & tb_2 & tb_3 & \dots & tb_n& \\
b_1 & a_2 + tb_2 & tb_3 & \dots & tb_n \\ 
b_1 & tb_2 & a_3 + tb_3 & \dots & tb_n \\
\vdots & \vdots & \vdots& & \vdots \\
b_1  & tb_2 & tb_3 &\dots & a_n + tb_n 
\end{vmatrix}\\&+\begin{vmatrix} 
a_1+tb_1 & b_2 & tb_3 & \dots & tb_n& \\
tb_1 & b_2 & tb_3 & \dots & tb_n \\ 
tb_1 & b_2 & a_3 + tb_3 & \dots & tb_n \\
\vdots & \vdots & \vdots& & \vdots \\
tb_1  & b_2 & tb_3 &\dots & a_n + tb_n 
\end{vmatrix}\\
&+\dots\\
&+\begin{vmatrix} 
a_1+tb_1 & tb_2 & tb_3 & \dots & b_n& \\
tb_1 & a_2 + tb_2 & tb_3 & \dots & b_n \\ 
tb_1 & tb_2 & a_3 + tb_3 & \dots & b_n \\
\vdots & \vdots & \vdots& & \vdots \\
tb_1  & tb_2 & tb_3 &\dots & b_n 
\end{vmatrix}\end{array}$$
$$M_n(t)=M_n(0)+\int_0^tM_n'(u)\, du$$
A: $$\begin{array}{ll}
D_n&=\begin{vmatrix} 
a_n+b_n & b_{n-1} & b_{n-2} & \dots & b_1& \\
b_n & a_{n-1} + b_{n-1} & b_{n-2} & \dots & b_1 \\ 
b_n & b_{n-1} & a_{n-2} + b_{n-2} & \dots & b_1 \\
\vdots & \vdots & \vdots& & \vdots \\
b_n  & b_{n-1} & b_{n-2} &\dots & a_1 + b_1 
\end{vmatrix}\\
&=\begin{vmatrix} 
a_n & b_{n-1} & b_{n-2} & \dots & b_1& \\
0 & a_{n-1} + b_{n-1} & b_{n-2} & \dots & b_1 \\ 
0 & b_{n-1} & a_{n-2} + b_{n-2} & \dots & b_1 \\
\vdots & \vdots & \vdots& & \vdots \\
0  & b_{n-1} & b_{n-2} &\dots & a_1 + b_1 
\end{vmatrix}\\
&+\begin{vmatrix} 
b_n & b_{n-1} & b_{n-2} & \dots & b_1& \\
b_n & a_{n-1} + b_{n-1} & b_{n-2} & \dots & b_1 \\ 
b_n & b_{n-1} & a_{n-2} + b_{n-2} & \dots & b_1 \\
\vdots & \vdots & \vdots& & \vdots \\
b_n  & b_{n-1} & b_{n-2} &\dots & a_1 + b_1 
\end{vmatrix}\\
&=a_nD_{n-1}+b_n\begin{vmatrix} 
1 & b_{n-1} & b_{n-2} & \dots & b_1& \\
1 & a_{n-1} + b_{n-1} & b_{n-2} & \dots & b_1 \\ 
1 & b_{n-1} & a_{n-2} + b_{n-2} & \dots & b_1 \\
\vdots & \vdots & \vdots& & \vdots \\
1  & b_{n-1} & b_{n-2} &\dots & a_1 + b_1 
\end{vmatrix}\\
&=a_nD_{n-1}+b_n\begin{vmatrix} 
1 & 0 & 0 & \dots & 0 & \\
1 & a_{n-1} & 0 & \dots & 0 \\ 
1 & 0 & a_{n-2} & \dots & 0 \\
\vdots & \vdots & \vdots& & \vdots \\
1  & 0 & 0 &\dots & a_1
\end{vmatrix}\\
&= a_n D_{n-1}+b_n\prod_{k=1}^{n-1}a_k
\end{array}$$
