I get a wrong determinant - why? I'm trying to calculate the following determinant:
$$\begin{vmatrix}
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & x & a_2 & \dots & a_n \\
a_0 & a_1 & x & \dots & a_n \\
\dots & \dots & \dots & \dots & \dots \\
a_0 & a_1 & a_2 & \dots & x
\end{vmatrix} = $$ $$ = \begin{vmatrix}
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & a_1 & a_2 & \dots & a_n \\
\dots & \dots & \dots & \dots & \dots \\
a_0 & a_1 & a_2 & \dots & a_n
\end{vmatrix} + \begin{vmatrix}
0 & 0 & 0 & \dots & 0 \\
0 & x - a_1 & 0 & \dots & 0 \\
0 & 0 & x - a_2 & \dots & 0 \\
\dots & \dots & \dots & \dots & \dots \\
0 & 0 & 0 & \dots & x-a_n
\end{vmatrix} = 0 + 0 = 0 $$
Still, experimental results contradict, since for one example I get a non-zero determinant.
What am I doing wrong?
 A: $\text{Det}(AB) = \text{Det}(A)\text{Det}(B)$, but in general, $\text{Det}(A+B)$ is not equal to $\text{Det}(A) +\text{Det}(B)$. It looks like you have used this wrong formula $\text{Det}(A+B) = \text{Det}(A) +\text{Det}(B)$ for your first equality.
A: Determinant is a multilinear map so the linearity works in a different way. Operation that you made corresponds to linearity in a sense
$\det(A+B) = \det(A) + \det(B)$. This is true in general only for 1x1 matrices. The correct way to use the linearity of determinant is
$$
\det(x_1 + \alpha y_1,x_2,\cdots,x_n) = \det(x_1,x_2,\cdots,x_n) + \alpha \det(y_1,x_2,\cdots,x_n)
$$
where $x_1,\cdots,x_n,y_1$ are vectors of rows/columns of the matrix. So in your case the correct manipulation is
$$
\begin{vmatrix}
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & x & a_2 & \dots & a_n \\
a_0 & a_1 & x & \dots & a_n \\
\dots & \dots & \dots & \dots & \dots \\
a_0 & a_1 & a_2 & \dots & x
\end{vmatrix} = \begin{vmatrix}
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & x - a_1 + a_1 & a_2 & \dots & a_n \\
a_0 & a_1 & x & \dots & a_n \\
\dots & \dots & \dots & \dots & \dots \\
a_0 & a_1 & a_2 & \dots & x
\end{vmatrix} =
\begin{vmatrix}
a_0 & 0 & a_2 & \dots & a_n \\
a_0 & x - a_1 & a_2 & \dots & a_n \\
a_0 & 0 & x & \dots & a_n \\
\dots & \dots & \dots & \dots & \dots \\
a_0 & 0 & a_2 & \dots & x
\end{vmatrix} +
\begin{vmatrix}
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & a_1 & x & \dots & a_n \\
\dots & \dots & \dots & \dots & \dots \\
a_0 & a_1 & a_2 & \dots & x
\end{vmatrix}
$$
A: I assume you actually want to know what the determinant is. For a 2 by 2 we have,
$$(1) \quad 
        \begin{vmatrix}
        a_0 & a_1 \\
        a_0 & x \\
        \end{vmatrix}
=a_0 \cdot (x-a_1)$$
For a 3 by 3, we have,
$$
        \begin{vmatrix}
        a_0 & a_1 & a_2 \\
        a_0 & x & a_2 \\
        a_0 & a_1 & x \\
        \end{vmatrix}
$$
Since the upper right 2 by 2 is already known, we can evaluate along the bottom row. If we also note that the expansion along the bottom $a_1$ is zero, we get,
$$(2) \quad
        \begin{vmatrix}
        a_0 & a_1 & a_2 \\
        a_0 & x & a_2 \\
        a_0 & a_1 & x \\
        \end{vmatrix}
=a_0 \cdot (x-a_1) \cdot (x-a_2)$$
In fact, we now know that the determinant $D_n$ for a n by n matrix of this form obeys,
$$(3) \quad D_{n+1}=D_n \cdot (x-a_{n-1})$$
Which implies using $D_1=a_0$, that
$$(4) \quad D_{n}=a_0 \cdot \prod_{k=1}^{n-1} (x-a_k)$$
I'm more physically oriented, so I'd just take this result and calculate the corresponding path integral. However, if you wish, I leave it as an exercise to rigorously prove this using induction.
