非齐次线性方程组不同解向量的系数相加等于 1 时，相加所得的向量也是该方程的解

一、题目

[A]. $k_{1}$ $+$ $k_{2}$ $+$ $k_{3}$ $=$ $1$

[B]. $k_{1}$ $+$ $k_{2}$ $+$ $k_{3}$ $=$ $3$

[C]. $k_{1}$ $\times$ $k_{2}$ $\times$ $k_{3}$ $=$ $1$

[D]. $k_{1}$ $=$ $1$ 且 $k_{2}$ $=$ $1$ 且 $k_{3}$ $=$ $1$

难度评级：

二、解析

由题可知，对于非齐次线性方程组 $\boldsymbol{A x}$ $=$ $b$ 而言，我们有：

$$
\textcolor{yellow}{
\begin{aligned}
\boldsymbol{A} \boldsymbol{\eta}_{1} & = b \\
\boldsymbol{A} \boldsymbol{\eta}_{2} & = b \\
\boldsymbol{A} \boldsymbol{\eta}_{3} & = b
\end{aligned}
}
$$

于是：

$$
\textcolor{yellow}{
\begin{aligned}
\boldsymbol{A} (\textcolor{magenta}{k_{1}} \boldsymbol{\eta}_{1} ) & = \textcolor{magenta}{k_{1}} b \\
\boldsymbol{A} (\textcolor{magenta}{k_{2}} \boldsymbol{\eta}_{2} ) & = \textcolor{magenta}{k_{2}} b \\
\boldsymbol{A} (\textcolor{magenta}{k_{3}} \boldsymbol{\eta}_{3} ) & = \textcolor{magenta}{k_{3}} b
\end{aligned}
}
$$

进而：

$$
\textcolor{yellow}{
\begin{aligned}
\boldsymbol{A} ( & \textcolor{magenta}{k_{1}} \boldsymbol{\eta}_{1} + \textcolor{magenta}{k_{2}} \boldsymbol{\eta}_{2} + \textcolor{magenta}{k_{3}} \boldsymbol{\eta}_{3}) \\
= & \textcolor{magenta}{k_{1}} b + \textcolor{magenta}{k_{2}} b + \textcolor{magenta}{k_{3}} b \\
= & (\textcolor{magenta}{k_{1} + k_{2} + k_{3}}) b
\end{aligned}
}
$$

因此，只有当：

$$
\textcolor{magenta}{k_{1} + k_{2} + k_{3}} = \textcolor{springgreen}{1}
$$

才会有：

$$
\textcolor{yellow}{
\begin{aligned}
\boldsymbol{A} (\textcolor{magenta}{k_{1}} \boldsymbol{\eta}_{1} + \textcolor{magenta}{k_{2}} \boldsymbol{\eta}_{2} + \textcolor{magenta}{k_{3}} \boldsymbol{\eta}_{3}) & = \textcolor{springgreen}{1} \times b \\
& = b
\end{aligned}
}
$$

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