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Need some help with Matrix!

For any arbitrary matrix,

\(\left[ \begin{matrix} { a }_{ n+1 } \\ { b }_{ n+1 } \end{matrix} \right] \quad =\quad \begin{bmatrix} \sqrt { 3 } +1 & \quad 1-\sqrt { 3 } \\ \sqrt { 3 } -1 & 1+\sqrt { 3 } \end{bmatrix}\quad \left[ \begin{matrix} { a }_{ n } \\ { b }_{ n } \end{matrix} \right] \)

where \(n\in N\) Also, \({ a }_{ n }={ b }_{ n }=1\)

Find \({ a }_{ 22 }\)

P.S. Thanks to my friend Shantam, for the question!!

Note by Abhineet Nayyar
2 years, 8 months ago

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Is the answer\( (2\sqrt{2})^{21} \) ?

Sudeep Salgia - 2 years, 8 months ago

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Yes I also getting same ..... what's ur method ?

I did in this way......

\(\\ \begin{cases} \vec { { V }_{ n } } ={ a }_{ n }i+{ b }_{ n }j \\ \vec { { V }_{ n+1 } } ={ a }_{ n+1 }i+{ b }_{ n+1 }j \end{cases}\\ \)

Now using matrix manuipaltion method that axis is rotted by an angle of 15 degrre ...

\(\displaystyle{\vec { { V }_{ n+1 } } =(2\sqrt { 2 } )\vec { { V }_{ n } } \\ \vec { { V }_{ 1 } } =i+j\quad (\because \quad a_{ 1 }=b_{ 1 }=1)\\ \\ \vec { { V }_{ 22 } } =(2\sqrt { 2 } )\vec { { V }_{ 21 } } ={ \left( 2\sqrt { 2 } \right) }^{ 21 }\vec { { V }_{ 1 } } }\)

Now simply comparing coffecients of An and Bn we get answer.....

Is it correct ....? @Sudeep Salgia

Karan Shekhawat - 2 years, 8 months ago

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Yep. There is a small typo in the first equation. It should be \( \displaystyle \vec{ V_{n+1} } = ( 2 \sqrt{2} ) e^{\frac{i \pi }{12}} \vec{V_n } \) And the net rotation is \( 315^{ \circ } \)with the initial angle being \( 45^{\circ } \) making the angle \( 360^{\circ} \). Hence \( a_n = (2 \sqrt{2} )^{21} \cos 360^{\circ} \) and \( b_n = (2 \sqrt{2} )^{21} \sin 360^{\circ} =0 \)

Sudeep Salgia - 2 years, 8 months ago

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@Sudeep Salgia Yes thanks... I missed in typing ... And Abhineet I don't think there is another method 0 ... if fact This question is designed using the concept of Matrix Transformation method ... which I learnt from here brilliant in a question This in which deepanshu gupta posted solution by using this concept ....!

So I think this question is specially designed for this concept... May be possible some other method but I think they will surly not an elegant one (at least time consuming) ... But I don't have any ideas about them yet...

Karan Shekhawat - 2 years, 8 months ago

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@Karan Shekhawat @Karan Shekhawat Thanks, actually i found it in an MTG magazine for engineering, well, my friend did...So, it is possible that the question may be from vectors, and not matrices. Anyways, Solutions are welcome. So, let's see:)

Abhineet Nayyar - 2 years, 8 months ago

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@Karan Shekhawat Also I like your status very much ..... :)

Karan Shekhawat - 2 years, 8 months ago

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@Karan Shekhawat Haha...Thank You!!:D:D

Abhineet Nayyar - 2 years, 8 months ago

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Thanks guys,but is there a method to solve it by Matrix algebra and not Vectors...?

Abhineet Nayyar - 2 years, 8 months ago

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@Abhineet Nayyar I have a method which is not so intuitive and I will post it as soon as I get some time.

Sudeep Salgia - 2 years, 8 months ago

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@Sudeep Salgia Thanks @Sudeep Salgia and @Karan Shekhawat for your help:)

Abhineet Nayyar - 2 years, 8 months ago

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@Sudeep Salgia Thanks ... we are egarly waiting ....

Karan Shekhawat - 2 years, 8 months ago

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