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第2042题：特殊的拉普拉斯展开



特殊“下三角行列式”


∣a11⋯a1nc11⋯c1m⋮⋮⋮⋮an1⋯anncn1⋯cnm0⋯0b11⋯b1m⋮⋮⋮⋮0⋯0bm1⋯bmm∣\begin{vmatrix} a_{11} & \cdots & a_{1n} & c_{11} & \cdots & c_{1m} \\ \vdots & & \vdots & \vdots & & \vdots \\ a_{n1} & \cdots & a_{nn} & c_{n1} & \cdots & c_{nm} \\ 0 & \cdots & 0 & b_{11} & \cdots & b_{1m} \\ \vdots & & \vdots & \vdots & & \vdots \\ 0 & \cdots & 0 & b_{m1} & \cdots & b_{mm} \end{vmatrix}


及特殊“上三角行列式”


∣a11⋯a1n0⋯0⋮⋮⋮⋮an1⋯ann0⋯0c11⋯c1nb11⋯b1m⋮⋮⋮⋮cm1⋯cmnbm1⋯bmm∣\begin{vmatrix} a_{11} & \cdots & a_{1n} & 0 & \cdots & 0 \\ \vdots & & \vdots & \vdots & & \vdots \\ a_{n1} & \cdots & a_{nn} & 0 & \cdots & 0 \\ c_{11} & \cdots & c_{1n} & b_{11} & \cdots & b_{1m} \\ \vdots & & \vdots & \vdots & & \vdots \\ c_{m1} & \cdots & c_{mn} & b_{m1} & \cdots & b_{mm} \end{vmatrix}



可展开为



A. ∣a11⋯a1n⋮⋮an1⋯ann∣∣c11⋯c1m⋮⋮cn1⋯cnm∣\begin{vmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{n1} & \cdots & a_{nn} \end{vmatrix} \begin{vmatrix} c_{11} & \cdots & c_{1m} \\ \vdots & & \vdots \\ c_{n1} & \cdots & c_{nm} \end{vmatrix} 



B.∣a11⋯a1n⋮⋮an1⋯ann∣∣b11⋯b1m⋮⋮bm1⋯bmm∣\begin{vmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{n1} & \cdots & a_{nn} \end{vmatrix} \begin{vmatrix} b_{11} & \cdots & b_{1m} \\ \vdots & & \vdots \\ b_{m1} & \cdots & b_{mm} \end{vmatrix} 



C.∣c11⋯c1m⋮⋮cn1⋯cnm∣∣b11⋯b1m⋮⋮bm1⋯bmm∣ \begin{vmatrix} c_{11} & \cdots & c_{1m} \\ \vdots & & \vdots \\ c_{n1} & \cdots & c_{nm} \end{vmatrix} \begin{vmatrix} b_{11} & \cdots & b_{1m} \\ \vdots & & \vdots \\ b_{m1} & \cdots & b_{mm} \end{vmatrix} 



D. ∣a11⋯a1m⋮⋮an1⋯anm∣∣b11⋯b1m⋮⋮bn1⋯bnm∣\begin{vmatrix} a_{11} & \cdots & a_{1m} \\ \vdots & & \vdots \\ a_{n1} & \cdots & a_{nm} \end{vmatrix} \begin{vmatrix} b_{11} & \cdots & b_{1m} \\ \vdots & & \vdots \\ b_{n1} & \cdots & b_{nm} \end{vmatrix} 

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