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Category: Algebra

let-A-0-m-m-2-1-m-0-m-1-m-2-1-m-0-A-M-3-R-and-m-not-0-1-find-relation-betwen-I-3-A-and-A-2-2-is-A-inversible-dete

Question Number 50366 by prof Abdo imad last updated on 16/Dec/18 $${let}\:\:{A}\:=\begin{pmatrix}{\mathrm{0}\:\:\:\:\:{m}\:\:\:\:\:\:{m}^{\mathrm{2}} }\\{\frac{\mathrm{1}}{{m}}\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:{m}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{1}}{{m}^{\mathrm{2}} }\:\:\:\:\frac{\mathrm{1}}{{m}}\:\:\:\:\:\:\mathrm{0}\:\:\:\right) \\ $$$${A}\:\in\:{M}_{\mathrm{3}} \left({R}\right)\:\:{and}\:{m}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{relation}\:{betwen}\:{I}_{\mathrm{3}} ,\:{A}\:{and}\:{A}^{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{is}\:{A}\:{inversible}\:\:\:.{determine}\:{A}^{−\mathrm{1}} \:{in}\:{case}\:{of}\:{exist}…

If-a-1-b-b-1-c-c-1-a-then-prove-that-abc-1-a-b-c-

Question Number 181438 by Agnibhoo98 last updated on 26/Nov/22 $$\mathrm{If}\:{a}\:+\:\frac{\mathrm{1}}{{b}}\:=\:{b}\:+\:\frac{\mathrm{1}}{{c}}\:=\:{c}\:+\:\frac{\mathrm{1}}{{a}}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${abc}\:=\:\pm\mathrm{1}.\:\:\:{a}\:\neq\:{b}\:\neq\:{c} \\ $$ Commented by mr W last updated on 26/Nov/22 $${it}'{s}\:{not}\:{true}.\:{e}.{g}.\:{you}\:{can}\:{take}\: \\ $$$${a}={b}={c}=\mathrm{2},\:{and}\:{get}\:{abc}=\mathrm{8}.…

let-J-1-1-1-1-1-1-1-1-1-element-of-M-3-R-find-J-n-

Question Number 50365 by prof Abdo imad last updated on 16/Dec/18 $${let}\:{J}\:=\begin{bmatrix}{\mathrm{1}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{bmatrix}{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\\{}\end{bmatrix} \\ $$$${element}\:{of}\:{M}_{\mathrm{3}} \left({R}\right) \\ $$$${find}\:{J}^{{n}} \\ $$$$ \\ $$ Terms of…