Question Number 175758 by Stephan last updated on 06/Sep/22
Answered by Jamshidbek last updated on 06/Sep/22
![Solution. we know tr(XY)=tr(YX) ⇒ tr(XY)=1 U+V=1 [1] det(XY)=det(X)det(Y) det(YX)=det(X)det(Y) det(XY)=−6 ⇒ det(YX)=−6 ⇒ UV=−6 [2] From [1] and [2] U=−2 ,V=3 or U=3 ,V=−2](https://www.tinkutara.com/question/Q175763.png)
$$\mathrm{Solution}. \\ $$$$\mathrm{we}\:\mathrm{know}\:\mathrm{tr}\left(\mathrm{XY}\right)=\mathrm{tr}\left(\mathrm{YX}\right)\:\Rightarrow\:\mathrm{tr}\left(\mathrm{XY}\right)=\mathrm{1} \\ $$$$\mathrm{U}+\mathrm{V}=\mathrm{1}\:\left[\mathrm{1}\right] \\ $$$$\mathrm{det}\left(\mathrm{XY}\right)=\mathrm{det}\left(\mathrm{X}\right)\mathrm{det}\left(\mathrm{Y}\right) \\ $$$$\mathrm{det}\left(\mathrm{YX}\right)=\mathrm{det}\left(\mathrm{X}\right)\mathrm{det}\left(\mathrm{Y}\right) \\ $$$$\mathrm{det}\left(\mathrm{XY}\right)=−\mathrm{6}\:\Rightarrow\:\mathrm{det}\left(\mathrm{YX}\right)=−\mathrm{6}\:\Rightarrow\:\mathrm{UV}=−\mathrm{6}\:\left[\mathrm{2}\right] \\ $$$$\mathrm{From}\:\left[\mathrm{1}\right]\:\mathrm{and}\:\left[\mathrm{2}\right] \\ $$$$\mathrm{U}=−\mathrm{2}\:,\mathrm{V}=\mathrm{3}\:\mathrm{or}\:\mathrm{U}=\mathrm{3}\:,\mathrm{V}=−\mathrm{2} \\ $$