Question Number 55844 by gunawan last updated on 05/Mar/19
$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{an}\:\mathrm{involutory}\:\mathrm{matrix},\:\mathrm{then}\: \\ $$$$\left({I}+{A}\right)\left({I}−{A}\right)=\mathrm{0}. \\ $$
Answered by 121194 last updated on 05/Mar/19
$$\mathrm{a}\:\mathrm{involutory}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{that}\:\mathrm{is}\:\mathrm{it}\:\mathrm{own}\:\mathrm{inverse} \\ $$$${A}^{\mathrm{2}} ={I} \\ $$$$\left({I}+{A}\right)\left({I}−{A}\right)={I}^{\mathrm{2}} −{IA}+{AI}−{A}^{\mathrm{2}} \\ $$$$={I}^{\mathrm{2}} −{I} \\ $$$$=\mathrm{0} \\ $$