每週問題 September 22, 2014

這是從特徵多項式判斷行列式的問題。

Find \det A given that A has p(t) as its characteristic polynomial.

(a) p(t)=t^3-2t^2+t+5
(b) p(t)=t^4-t^3+7

 
參考解答:

給定的特徵多項式的領先係數為 1,我們定義特徵多項式為 p(t)=\det(tI-A)

(a) 因為 p(t) 為一三次多項式,可知 A 是一 3\times 3 階矩陣,故 p(0)=5=\det(-A)=(-1)^3(\det A)=-\det A,所求為 \det A=-5

(b) 因為 p(t) 為一四次多項式,可知 A 是一 4\times 4 階矩陣,故 p(0)=7=\det(-A)=(-1)^4(\det A)=\det A

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