设有齐次线性方程组
$$
\left\{\begin{array}{l}
(1+a) x_{1}+x_{2}+\cdots+x_{n}=0, \\
2 x_{1}+(2+a) x_{2}+\cdots+2 x_{n}=0, \\
\cdots \cdots \\
n x_{1}+n x_{2}+\cdots+(n+a) x_{n}=0,
\end{array} \quad(n \geqslant 2),\right.
$$
试问 $a$ 取何值时, 该方程组有非零解, 并求出其通解.
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$