考研线性代数：行列式部分初级专项练习题（2024 年） - 第10页共13页

题目 10

$D = | \begin{array}{lll} b + c & c + a & a + b \\ b_{1} + c_{1} & c_{1} + a_{1} & a_{1} + b_{1} \\ b_{2} + c_{2} & c_{2} + a_{2} & a_{2} + b_{2} \end{array} | = ?$

解析 10

$D = | \begin{array}{lll} b + c & c + a & a + b \\ b_{1} + c_{1} & c_{1} + a_{1} & a_{1} + b_{1} \\ b_{2} + c_{2} & c_{2} + a_{2} & a_{2} + b_{2} \end{array} | =$

$| \begin{array}{lll} b & c + a & a + b \\ b_{1} & c_{1} + a_{1} & a_{1} + b_{1} \\ b_{2} & c_{2} + a_{2} & a_{2} + b_{2} \end{array} | + | \begin{array}{ccc} c & c + a & a + b \\ c_{1} & c_{1} + a_{1} & a_{1} + b_{1} \\ c_{2} & c_{2} + a_{2} & a_{2} + b_{2} \end{array} | =$

$| \begin{array}{ccc} b & c + a & a \\ b_{1} & c_{1} + a_{1} & a_{1} \\ b_{2} & c_{2} + a_{2} & a_{2} \end{array} | + | \begin{array}{ccc} c & a & a + b \\ c_{1} & a_{1} & a_{1} + b_{1} \\ c_{2} & a_{2} & a_{2} + b_{2} \end{array} | =$

$| \begin{array}{lll} b & c & a \\ b_{1} & c_{1} & a_{1} \\ b_{2} & c_{2} & a_{2} \end{array} | + | \begin{array}{ccc} c & a & b \\ c_{1} & a_{1} & b_{1} \\ c_{2} & a_{2} & b_{2} \end{array} | =$

$| \begin{array}{lll} a & b & c \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} | + | \begin{array}{lll} a & b & c \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} | =$

$2 | \begin{array}{lll} a & b & c \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} | .$

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