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Question Number 204957 by CrispyXYZ last updated on 03/Mar/24

$$2\times 2\mathrm{matrix}\mathbf{A}\mathrm{and}\mathbf{B}\mathrm{satisfy}\mathrm{that}$$$$\mathbf{AB}+\mathbf{A}=\mathbf{BA}+\mathbf{B}.$$$$\mathrm{Prove}\mathrm{that}{(\mathbf{A}-\mathbf{B})}^2=\mathbf{O}.$$

Answered by Rajpurohith last updated on 04/Mar/24

$$Neednotholdtrue!$$$$takeA=-IandB=Isatisfythegivencondition$$$$yet{(A-B)}^2\ne 0$$

Commented by MM42 last updated on 04/Mar/24

$$-2I\ne 0$$

Answered by witcher3 last updated on 04/Mar/24

$$\mathrm{A}=\left(\begin{array}{c}\mathrm{a}\mathrm{b}\\ \mathrm{c}\mathrm{d}\end{array}\right);\mathrm{B}=\left(\begin{array}{c}\mathrm{e}\mathrm{f}\\ \mathrm{g}\mathrm{h}\end{array}\right)$$$$\mathrm{AB}+\mathrm{A}=\left(\begin{array}{c}\mathrm{ae}+\mathrm{bg}+\mathrm{a}\mathrm{af}+\mathrm{bh}+\mathrm{b}\\ \mathrm{ce}+\mathrm{dg}+\mathrm{c}\mathrm{cf}+\mathrm{dh}+\mathrm{d}\end{array}\right)$$$$\mathrm{BA}+\mathrm{B}=\left(\begin{array}{c}\mathrm{ae}+\mathrm{fc}+\mathrm{e}\mathrm{eb}+\mathrm{fd}+\mathrm{f}\\ \mathrm{ga}+\mathrm{hc}+\mathrm{g}\mathrm{gb}+\mathrm{hd}+\mathrm{h}\end{array}\right)$$$$\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{Tr}\left(\mathrm{AB}\right)=\mathrm{Tr}\left(\mathrm{BA}\right)$$$$\mathrm{Tr}(\mathrm{AB}+\mathrm{A})=\mathrm{Tr}(\mathrm{BA}+\mathrm{B})\Rightarrow \mathrm{Tr}\left(\mathrm{AB}\right)+\mathrm{Tr}\left(\mathrm{A}\right)=\mathrm{Tr}\left(\mathrm{BA}\right)+\mathrm{Tr}\left(\mathrm{B}\right)$$$$\Rightarrow \mathrm{Tr}\left(\mathrm{A}\right)=\mathrm{Tr}\left(\mathrm{B}\right)\mathrm{a}+\mathrm{d}=\mathrm{e}+\mathrm{h}$$$$\det (\mathrm{AB}+\mathrm{A})=(\mathrm{ae}+\mathrm{bg}+\mathrm{a})(\mathrm{cf}+\mathrm{dh}+\mathrm{d})-(\mathrm{af}+\mathrm{bh}+\mathrm{b})(\mathrm{ce}+\mathrm{dg}+\mathrm{c})$$$$=\det (\mathrm{BA}+\mathrm{B})=(\mathrm{ae}+\mathrm{fc}+\mathrm{e})(\mathrm{gb}+\mathrm{hd}+\mathrm{h})-(\mathrm{ga}+\mathrm{hc}+\mathrm{g})(\mathrm{eb}+\mathrm{fd}+\mathrm{f})$$$$(\mathrm{a}-\mathrm{e})(\mathrm{d}-\mathrm{h})-(\mathrm{b}-\mathrm{f})(\mathrm{c}-\mathrm{g})=0$$$$\Rightarrow \det (\mathrm{A}-\mathrm{B})=0$$$$\mathrm{X}=\mathrm{A}-\mathrm{B}\Rightarrow \mathrm{Tr}\left(\mathrm{X}\right)=0,\det \left(\mathrm{X}\right)=0$$$$\mathrm{we}{\mathrm{knowX}}^2-\mathrm{tr}\left(\mathrm{X}\right).\mathrm{X}+\det \left(\mathrm{X}\right)=0\Rightarrow {\mathrm{X}}^2=0$$$${(\mathrm{A}-\mathrm{B})}^2={\mathrm{O}}_{2\ast 2}$$$$$$$$$$

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