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Question Number 12044 by 7991 last updated on 10/Apr/17
$${A}\in{M}_{{n}×{n}} \\ $$$${A}^{\mathrm{2}} ={A} \\ $$$$\left({I}+{A}\right)^{−\mathrm{1}} =....??? \\ $$
Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 10/Apr/17
$${X}={I}+{A} \\ $$$${Y}={I}−{A} \\ $$$${Y}={I}^{\mathrm{2}} −{A}^{\mathrm{2}} =\left({I}+{A}\right)\left({I}−{A}\right)={XY} \\ $$$${X}^{−\mathrm{1}} .{Y}={X}^{−\mathrm{1}} .{XY}={I}.{Y}={Y} \\ $$$${X}^{−\mathrm{1}} .{Y}.{Y}^{−\mathrm{1}} ={Y}.{Y}^{−\mathrm{1}} ={I}\Rightarrow{X}^{−\mathrm{1}} =\left({I}+{A}\right)^{−\mathrm{1}} ={I} \\ $$