每週問題 May 15, 2017

反對稱矩陣的伴隨矩陣 (adjugate) 是對稱或反對稱矩陣。

Let A be an n\times n skew-symmetric matrix. Prove that \hbox{adj}A is a symmetric matrix for odd n and a skew-symmetric matrix for even n.

 
參考解答:

假設 A 是一個反對稱矩陣,A^T=-A,使用伴隨矩陣性質 \hbox{adj}(A^T)=(\hbox{adj}A)^T\hbox{adj}(-A)=(-1)^{n-1}\hbox{adj}A

(\hbox{adj}A)^T=\hbox{adj}(A^T)=\hbox{adj}(-A)=(-1)^{n-1}\hbox{adj}A

n 為奇數,則 (\hbox{adj}A)^T=\hbox{adj}A;若 n 為偶數,則 (\hbox{adj}A)^T=-\hbox{adj}A

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