每週問題 October 5, 2015

證明 Jacobi 行列式恆等式。

Let A=\begin{bmatrix}  B&C\\  D&E  \end{bmatrix} be an n\times n nonsingular matrix, where B is k\times k. Denote the adjugate of A by \hbox{adj}\,A=\begin{bmatrix}  P&Q\\  R&S  \end{bmatrix}, where P is k\times k. Prove the Jacobi identity

\det P=(\det A)^{k-1}(\det E).

 
參考解答:

使用伴隨矩陣恆等式 (\hbox{adj}\,A)A=(\det A)I_n,即

\begin{bmatrix}  P&Q\\  R&S  \end{bmatrix}\begin{bmatrix}  B&C\\  D&E  \end{bmatrix}=\begin{bmatrix}  (\det A)I_k&0\\  0&(\det A)I_{n-k}  \end{bmatrix}

可得

\begin{bmatrix}  P&Q\\  0&I_{n-k}  \end{bmatrix}A=\begin{bmatrix}  P&Q\\  0&I_{n-k}  \end{bmatrix}\begin{bmatrix}  B&C\\  D&E  \end{bmatrix}=\begin{bmatrix}  (\det A)I_k&0\\  D&E  \end{bmatrix}

等號兩邊取行列式,

(\det P)(\det A)=\left(\det((\det A)I_k)\right)(\det E)=(\det A)^k(\det E)

因為 \det A\neq 0\det P=(\det A)^{k-1}(\det E)

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