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Question Number 175758 by Stephan last updated on 06/Sep/22

Answered by Jamshidbek last updated on 06/Sep/22

$$\mathrm{Solution}.$$$$\mathrm{we}\mathrm{know}\mathrm{tr}\left(\mathrm{XY}\right)=\mathrm{tr}\left(\mathrm{YX}\right)\Rightarrow \mathrm{tr}\left(\mathrm{XY}\right)=1$$$$\mathrm{U}+\mathrm{V}=1\left[1\right]$$$$\det \left(\mathrm{XY}\right)=\det \left(\mathrm{X}\right)\det \left(\mathrm{Y}\right)$$$$\det \left(\mathrm{YX}\right)=\det \left(\mathrm{X}\right)\det \left(\mathrm{Y}\right)$$$$\det \left(\mathrm{XY}\right)=-6\Rightarrow \det \left(\mathrm{YX}\right)=-6\Rightarrow \mathrm{UV}=-6\left[2\right]$$$$\mathrm{From}\left[1\right]\mathrm{and}\left[2\right]$$$$\mathrm{U}=-2,\mathrm{V}=3\mathrm{or}\mathrm{U}=3,\mathrm{V}=-2$$

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