行列式02---余子式与代数余子式——行列式按一行(列)展开
Posted 炫云云
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理解余子式、代数余子式的概念，学会行列式的按行（列）展开法，从而掌握行列式计算的一种技巧.
引入
三级行列式展开式:
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\\begin{aligned} D_{3}=\\left|\\begin{array}{lll} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\end{array}\\right| &=a_{11} a_{22} a_{33}+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{32}-a_{11} a_{23} a_{32}-a_{12} a_{21} a_{33}-a_{13} a_{22} a_{31} \\\\ &=a_{11}\\left(a_{22} a_{33}-a_{23} a_{32}\\right)+a_{12}\\left(a_{23} a_{31}-a_{21} a_{33}\\right)+a_{13}\\left(a_{21} a_{32}-a_{22} a_{31}\\right) \\\\ &=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \\end{aligned}
D3=∣∣∣∣∣∣a11a21a31a12a22a32a13a23a33∣∣∣∣∣∣=a11a22a33+a12a23a31+a13a21a32−a11a23a32−a12a21a33−a13a22a31=a11(a22a33−a23a32)+a12(a23a31−a21a33)+a13(a21a32−a22a31)=a11A11+a12A12+a13A13
以此类推,
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D_{3}=a_{i 1} A_{i 1}+a_{i 2} A_{i 2}+a_{i 3} A_{i 3} .
D3=ai1Ai1+ai2Ai2+ai3Ai3.
再推,(按第
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D_{n}=\\left|\\begin{array}{cccc}a_{11} & a_{12} & \\cdots & a_{1 n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a_{i 1} & a_{i 2} & \\cdots & a_{i n} \\\\ \\vdots & \\vdots & & \\vdots \\\\ a_{n 1} & a_{n 2} & \\cdots & a_{n n}\\end{array}\\right|=a_{i 1} A_{i 1}+a_{i 2} A_{i 2}+\\cdots+a_{i n} A_{i n}, i=1,2, \\cdots, n\\tag{1}