Question Number 180188 by Shrinava last updated on 08/Nov/22
Answered by Rasheed.Sindhi last updated on 09/Nov/22
![Formulas: determinant ((( [(( l),m),((−m),l) ] [(( p),q),((−q),p) ]= [(( lp−mq),(lq+mp)),((−(lq+mp)),(lp−mq)) ]_ ^ ))) determinant (((( [(( l),m),((−m),l) ])^2 = [((l^2 −m^2 ),(2lm)),((−2lm),(l^2 −m^2 )) ]))) A= [(( a_1 ),b_1 ),((−b_1 ),a_1 ) ]= [(( 5),1),((−1),5) ] A^2 = [(( a_2 ),b_2 ),((−b_2 ),a_2 ) ]= [(( 24),(10)),((−10),(24)) ] (A^2 )^2 = [(( a_4 ),b_4 ),((−b_4 ),a_4 ) ]= [(( 476),(480)),((−480),(476)) ] A^4 ∙A= [(( 476),(480)),((−480),(476)) ] [(( 5),1),((−1),5) ] A^5 = [(( a_5 ),b_5 ),((−b_5 ),a_5 ) ]= [(( 1900),(2876)),((−2876),(1900)) ] (A^5 )^2 =( [(( 1900),(2876)),((−2876),(1900)) ])^2 A^(10) = [((−4661376),( 10928800)),((−10928800),(−4661376)) ] (A^(10) )^2 =( [((−4661376),( 10928800)),((−10928800),(−4661376)) ])^2 A^(20) = [(( a_(20) ),b_(20) ),((−b_(20) ),a_(20) ) ] = [((−97 710 243 226 624),(−101 886 492 057 600)),(( 101 886 492 057 600),( −97 710 243 226 624)) ] Ω=((a_(20) +ib_(20) ))^(1/3) =((−97 710 243 226 624−101 886 492 057 600i))^(1/3) =−((97 710 243 226 624+101 886 492 057 600i))^(1/3)](Q180228.png)
$$\mathcal{F}{ormulas}: \\ $$$$\begin{array}{|c|}{\:\:\begin{bmatrix}{\:\:\:\:\:{l}}&{{m}}\\{−{m}}&{{l}}\end{bmatrix}\begin{bmatrix}{\:\:\:\:{p}}&{{q}}\\{−{q}}&{{p}}\end{bmatrix}=\begin{bmatrix}{\:\:\:\:\:\:{lp}−{mq}}&{{lq}+{mp}}\\{−\left({lq}+{mp}\right)}&{{lp}−{mq}}\end{bmatrix}_{\:} ^{\:} }\\\hline\end{array}\: \\ $$$$\begin{array}{|c|}{\left(\begin{bmatrix}{\:\:\:\:\:{l}}&{{m}}\\{−{m}}&{{l}}\end{bmatrix}\right)^{\mathrm{2}} \:=\begin{bmatrix}{{l}^{\mathrm{2}} −{m}^{\mathrm{2}} }&{\mathrm{2}{lm}}\\{−\mathrm{2}{lm}}&{{l}^{\mathrm{2}} −{m}^{\mathrm{2}} }\end{bmatrix}}\\\hline\end{array} \\ $$$$ \\ $$$${A}=\begin{bmatrix}{\:\:\:\:{a}_{\mathrm{1}} }&{{b}_{\mathrm{1}} }\\{−{b}_{\mathrm{1}} }&{{a}_{\mathrm{1}} }\end{bmatrix}=\begin{bmatrix}{\:\:\:\:\mathrm{5}}&{\mathrm{1}}\\{−\mathrm{1}}&{\mathrm{5}}\end{bmatrix} \\ $$$${A}^{\mathrm{2}} =\begin{bmatrix}{\:\:\:\:{a}_{\mathrm{2}} }&{{b}_{\mathrm{2}} }\\{−{b}_{\mathrm{2}} }&{{a}_{\mathrm{2}} }\end{bmatrix}=\begin{bmatrix}{\:\:\:\:\mathrm{24}}&{\mathrm{10}}\\{−\mathrm{10}}&{\mathrm{24}}\end{bmatrix} \\ $$$$\left({A}^{\mathrm{2}} \right)^{\mathrm{2}} =\begin{bmatrix}{\:\:\:\:{a}_{\mathrm{4}} }&{{b}_{\mathrm{4}} }\\{−{b}_{\mathrm{4}} }&{{a}_{\mathrm{4}} }\end{bmatrix}=\begin{bmatrix}{\:\:\:\:\:\mathrm{476}}&{\mathrm{480}}\\{−\mathrm{480}}&{\mathrm{476}}\end{bmatrix} \\ $$$$\:\:\:{A}^{\mathrm{4}} \centerdot{A}=\begin{bmatrix}{\:\:\:\:\:\mathrm{476}}&{\mathrm{480}}\\{−\mathrm{480}}&{\mathrm{476}}\end{bmatrix}\begin{bmatrix}{\:\:\:\:\mathrm{5}}&{\mathrm{1}}\\{−\mathrm{1}}&{\mathrm{5}}\end{bmatrix} \\ $$$$\:\:{A}^{\mathrm{5}} =\begin{bmatrix}{\:\:\:\:{a}_{\mathrm{5}} }&{{b}_{\mathrm{5}} }\\{−{b}_{\mathrm{5}} }&{{a}_{\mathrm{5}} }\end{bmatrix}=\begin{bmatrix}{\:\:\:\:\mathrm{1900}}&{\mathrm{2876}}\\{−\mathrm{2876}}&{\mathrm{1900}}\end{bmatrix} \\ $$$$\:\left({A}^{\mathrm{5}} \right)^{\mathrm{2}} =\left(\begin{bmatrix}{\:\:\:\:\mathrm{1900}}&{\mathrm{2876}}\\{−\mathrm{2876}}&{\mathrm{1900}}\end{bmatrix}\right)^{\mathrm{2}} \\ $$$${A}^{\mathrm{10}} =\begin{bmatrix}{−\mathrm{4661376}}&{\:\:\:\:\:\mathrm{10928800}}\\{−\mathrm{10928800}}&{−\mathrm{4661376}}\end{bmatrix}\: \\ $$$$\left({A}^{\mathrm{10}} \right)^{\mathrm{2}} =\left(\begin{bmatrix}{−\mathrm{4661376}}&{\:\:\:\:\:\mathrm{10928800}}\\{−\mathrm{10928800}}&{−\mathrm{4661376}}\end{bmatrix}\right)^{\mathrm{2}} \\ $$$${A}^{\mathrm{20}} =\begin{bmatrix}{\:\:\:\:{a}_{\mathrm{20}} }&{{b}_{\mathrm{20}} }\\{−{b}_{\mathrm{20}} }&{{a}_{\mathrm{20}} }\end{bmatrix} \\ $$$$=\begin{bmatrix}{−\mathrm{97}\:\mathrm{710}\:\mathrm{243}\:\mathrm{226}\:\mathrm{624}}&{−\mathrm{101}\:\mathrm{886}\:\mathrm{492}\:\mathrm{057}\:\mathrm{600}}\\{\:\:\mathrm{101}\:\mathrm{886}\:\mathrm{492}\:\mathrm{057}\:\mathrm{600}}&{\:\:−\mathrm{97}\:\mathrm{710}\:\mathrm{243}\:\mathrm{226}\:\mathrm{624}}\end{bmatrix} \\ $$$$\Omega=\sqrt[{\mathrm{3}}]{{a}_{\mathrm{20}} +{ib}_{\mathrm{20}} }\: \\ $$$$=\sqrt[{\mathrm{3}}]{−\mathrm{97}\:\mathrm{710}\:\mathrm{243}\:\mathrm{226}\:\mathrm{624}−\mathrm{101}\:\mathrm{886}\:\mathrm{492}\:\mathrm{057}\:\mathrm{600}{i}}\: \\ $$$$=−\sqrt[{\mathrm{3}}]{\mathrm{97}\:\mathrm{710}\:\mathrm{243}\:\mathrm{226}\:\mathrm{624}+\mathrm{101}\:\mathrm{886}\:\mathrm{492}\:\mathrm{057}\:\mathrm{600}{i}}\: \\ $$
Commented by Shrinava last updated on 09/Nov/22
$$\mathrm{Thank}\:\mathrm{you},\:\mathrm{but}\:\mathrm{no}\:\mathrm{dear}\:\mathrm{professor} \\ $$
Commented by Rasheed.Sindhi last updated on 17/Nov/22
$$\mathcal{T}{hanks}\:{Shirinava},{please}\:{share} \\ $$$${your}\:{solution}\:{or}\:{at}\:{least}\:{answer}. \\ $$