考研线性代数：行列式部分初级专项练习题（2024 年） - 第12页共13页

题目 12

$D = | \begin{array}{cccc} 2 a & - 1 & 0 & 0 \\ a^{2} & 2 a & - 1 & 0 \\ 0 & a^{2} & 2 a & - 1 \\ 0 & 0 & a^{2} & 2 a \end{array} | =$

解析 12

$D = | \begin{array}{cccc} 2 a & - 1 & 0 & 0 \\ a^{2} & 2 a & - 1 & 0 \\ 0 & a^{2} & 2 a & - 1 \\ 0 & 0 & a^{2} & 2 a \end{array} | =$

$2 a | \begin{array}{ccc} 2 a & - 1 & 0 \\ a^{2} & 2 a & - 1 \\ 0 & a^{2} & 2 a \end{array} | + | \begin{array}{ccc} a^{2} & - 1 & 0 \\ 0 & 2 a & - 1 \\ 0 & a^{2} & 2 a \end{array} | =$

$2 a (2 a | \begin{array}{cc} 2 a & - 1 \\ a^{2} & 2 a \end{array} | + | \begin{array}{cc} a^{2} & - 1 \\ 0 & 2 a \end{array} |) +$

$a^{2} | \begin{array}{cc} 2 a & - 1 \\ a^{2} & 2 a \end{array} | + | \begin{array}{cc} 0 & - 1 \\ 0 & 2 a \end{array} | =$

$2 a [\begin{array}{ll} 2 a (4 a^{2} + a^{2}) + 2 a^{3} \end{array}] + a^{2} (4 a^{2} + a^{2}) + 0 =$

$2 a (10 a^{3} + 2 a^{3}) + 5 a^{4} =$

$24 a^{4} + 5 a^{4} = 29 a^{4} .$

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