考研线性代数：行列式部分初级专项练习题（2024 年） - 第11页共13页

题目 11

已知 $D = | \begin{array}{cccc} 2 & - 5 & 1 & 2 \\ - 3 & 7 & - 1 & 4 \\ 5 & - 9 & 2 & 7 \\ 4 & - 6 & 1 & 2 \end{array} |$ , 则 $D = ?$ , $M_{31} + M_{33} + M_{34} = ?$

解析 11

$D = | \begin{array}{cccc} 2 & - 5 & 1 & 2 \\ - 3 & 7 & - 1 & 4 \\ 5 & - 9 & 2 & 7 \\ 4 & - 6 & 1 & 2 \end{array} | =$

$| \begin{array}{cccc} - 2 & 1 & 1 & 0 \\ - 11 & 19 & - 1 & 6 \\ - 9 & 12 & 2 & 3 \\ 0 & 0 & 1 & 0 \end{array} | =$

$(- 1)^{4 + 3} \cdot | \begin{array}{ccc} - 2 & 1 & 0 \\ - 11 & 19 & 6 \\ - 9 & 12 & 3 \end{array} | =$

$- | \begin{array}{ccc} 0 & 1 & 0 \\ 27 & 19 & 6 \\ 15 & 12 & 3 \end{array} | =$

$| \begin{array}{cc} 27 & 6 \\ 15 & 3 \end{array} | =$

$27 \times 3 - 15 \times 6 =$

$(27 - 30) \times 3 = (- 3) \times 3 = - 9.$

接着：

$M_{31} + M_{33} + M_{34} =$

$(- 1)^{3 + 1} A_{31} + 0 \times A_{32} + (- 1)^{3 + 3} A_{33} + (- 1)^{3 + 4} A_{34} =$

$1 \times A_{31} + 0 \times A_{32} + 1 \times A_{33} + (- 1) \times A_{34} =$

$| \begin{array}{cccc} 2 & - 5 & 1 & 2 \\ - 3 & 7 & - 1 & 4 \\ 1 & 0 & 1 & - 1 \\ 4 & - 6 & 1 & 2 \end{array} | = | \begin{array}{cccc} 1 & - 5 & 1 & 3 \\ - 2 & 7 & - 1 & 3 \\ 0 & 0 & 1 & 0 \\ 3 & - 6 & 1 & 3 \end{array} | =$

$| \begin{array}{cccc} 1 & - 5 & 3 \\ - 2 & 7 & 3 \\ 3 & - 6 & 3 \end{array} | = | \begin{array}{ccc} 1 & - 5 & 3 \\ - 3 & 12 & 0 \\ 2 & - 1 & 0 \end{array} | =$

$3 \times | \begin{array}{ccc} - 3 & 12 \\ 2 & - 1 \end{array} | = 3 (3 - 24) = 3 \times (- 21) = - 63.$

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