设 $n(n \geqslant 2)$ 阶矩阵 $\boldsymbol{A}=\left(\begin{array}{ccccc}1 & 1 & 1 & \cdots & 1 \\ a & 1 & 0 & \cdots & 0 \\ a & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ a & 0 & 0 & \cdots & 1\end{array}\right), \boldsymbol{\beta}=\left(\begin{array}{c}b \\ 1 \\ 1 \\ \vdots \\ 1\end{array}\right)$, 求解方程组 $\boldsymbol{A} \boldsymbol{x}=\boldsymbol{\beta}$.
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