每週問題 January 19, 2015

如果 AA^{-1} 的所有元為整數,則 \det A 有甚麼性質?

Let A be an n\times n matrix. If all entries of A and A^{-1} are integers, show that \det(A^2)=1.

 
參考解答:

根據萊布尼茲行列式公式,AA^{-1} 的所有元皆為整數可推論 \det A\det(A^{-1})=(\det A)^{-1} 亦為整數,即知 \det A=\pm 1。所以,\det(A^2)=(\det A)^2=1

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