每週問題 January 12, 2015

這是二矩陣和的行列式推導問題。

Let A be an n\times n matrix. If A^TA=I and \det A<0, determine \det(A+I).

 
參考解答:

使用矩陣積的行列式可乘公式以及行列式的轉置不變性,可得

\displaystyle\begin{aligned}  \det(A+I)&=\det(A+A^TA)\\  &=\det((I+A^T)A)\\  &=\det(I+A^T)\det A\\  &=\det(I+A)\det A.  \end{aligned}

\det A<0,推知 \det (A+I)=0

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