If $$A=$$$$cos$$⁡θ−$$sin$$⁡θ$$sin$$⁡θ$$cos$$⁡θ, then the matrix A−$$50$$ when θ$$=$$π$$12$$, is equal to

Question

If $$A=$$$$cos$$⁡θ−$$sin$$⁡θ$$sin$$⁡θ$$cos$$⁡θ, then the matrix A−$$50$$ when θ$$=$$π$$12$$, is equal to

Infinity Learn · Accepted Answer

We  have $$__A=$$$$cos$$⁡θ−$$sin$$⁡θ$$sin$$⁡θ$$cos$$⁡θ∴   |A|$$=cos2$$θ$$+sin2$$θ$$=1$$ And adj $$A=$$$$cos$$⁡θ$$sin$$⁡θ−$$sin$$⁡θ$$cos$$⁡θ∵ If $$A=a$$    bc    d, then adj $$A=d$$−b−ca⇒   A−$$1=$$$$cos$$θ$$sin$$θ−$$sin$$θ$$cos$$θ     ∵A−$$1=adj$$  A|A| Note that, A−$$50=A$$−$$150$$ Now, A−$$2=A$$−$$1A$$−$$1$$⇒A−$$2=$$$$cos$$⁡θ$$sin$$⁡θ−$$sin$$⁡θ$$cos$$⁡θ$$cos$$⁡θ$$sin$$⁡θ−$$sin$$⁡θ$$cos$$⁡θ$$=cos2$$⁡θ−$$sin2$$⁡θ$$cos$$⁡θ$$sin$$⁡θ$$+$$$$sin$$⁡θ$$cos$$⁡θ−$$cos$$⁡θ$$sin$$⁡θ−$$cos$$⁡θ$$sin$$⁡θ−$$sin2$$⁡θ$$+cos2$$⁡θ$$=$$$$cos$$⁡$$2$$θ$$sin$$⁡$$2$$θ−$$sin$$⁡$$2$$θ$$cos$$⁡$$2$$θ Also, A−$$3=A$$−$$2A$$−$$1A$$−$$3=$$$$cos$$⁡$$2$$θ$$sin$$⁡$$2$$θ−$$sin$$⁡$$2$$θ$$cos$$⁡$$2$$θ$$cos$$⁡θ$$sin$$⁡θ−$$sin$$⁡θ$$cos$$⁡θ$$=$$$$cos$$⁡$$3$$θ$$sin$$⁡$$3$$θ−$$sin$$⁡$$3$$θ$$cos$$⁡$$3$$θ Similarly, A−$$50=$$$$cos$$⁡$$50$$θ$$sin$$⁡$$50$$θ−$$sin$$⁡$$50$$θ$$cos$$⁡$$50$$θ$$=$$$$cos$$⁡$$256$$$$π$$$$sin$$⁡$$256$$π−$$sin$$⁡$$256$$$$π$$$$cos$$⁡$$256$$π  when θ$$=$$π$$12=$$$$cos$$⁡π$$6sin$$⁡π$$6$$−$$sin$$⁡π$$6cos$$⁡π$$6$$∵$$cos$$⁡$$25$$$$π$$$$6=$$$$cos$$⁡$$4$$π$$+$$π$$6=$$$$cos$$⁡π$$6$$ and $$sin$$⁡$$25$$$$π$$$$6=$$$$sin$$⁡$$4$$π$$+$$π$$6=$$$$sin$$⁡π$$6=3212$$−$$1232$$

$If A = [\begin{array}{cc} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{array}], then the matrix A^{- 50} when θ = \frac{π}{12}, is equal to$

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If A=cos⁡θ−sin⁡θsin⁡θcos⁡θ, then the matrix A−50 when θ=π12, is equal to

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$If A = [\begin{array}{cc} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{array}], then the matrix A^{- 50} when θ = \frac{π}{12}, is equal to$