每週問題 May 15, 2017

Posted on 05/15/2017 by ccjou

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 與 \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。

若 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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