Let $\theta = {\pi \over 5}$ and $$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$.
If B = A + A4
, then det (B) :
Solution
$$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$
<br><br>A<sup>2</sup> = $$\left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$$$\left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$
<br><br>$\Rightarrow$ A<sup>2</sup> = $$\left[ {\matrix{
{\cos 2\theta } & {\sin 2\theta } \cr
{ - \sin 2\theta } & {\cos 2\theta } \cr
} } \right]$$
<br><br>Similarly, A<sup>n</sup> = $$\left[ {\matrix{
{\cos n\theta } & {\sin n\theta } \cr
{ - \sin n\theta } & {\cos n\theta } \cr
} } \right]$$
<br><br>$\therefore$ B = A + A<sup>4</sup>
<br><br>= $$\left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$ + $$\left[ {\matrix{
{\cos 4\theta } & {\sin 4\theta } \cr
{ - \sin 4\theta } & {\cos 4\theta } \cr
} } \right]$$
<br><br>= $$\left[ {\matrix{
{\cos 4\theta + \cos \theta } & {\sin 4\theta + \sin \theta } \cr
{ - \sin 4\theta - \sin \theta } & {\cos 4\theta + \cos \theta } \cr
} } \right]$$
<br><br>detB = (cos4$\theta$ + cos$\theta$)<sup>2</sup>
+ (sin4$\theta$ + sin$\theta$)<sup>2</sup>
<br><br>= cos<sup>2</sup>4$\theta$ + cos<sup>2</sup>$\theta$ + 2cos4$\theta$ cos$\theta$
<br>+ sin<sup>2</sup>4$\theta$ + sin2$\theta$ + 2sin4$\theta$ –sin$\theta$
<br><br>= 2 + 2 ( cos4$\theta$ cos$\theta$ + sin4$\theta$ sin$\theta$)
<br><br>$\Rightarrow$ detB = 2 + 2 cos3$\theta$
<br><br>at $\theta$ = ${\pi \over 5}$
<br><br>detB = 2 + 2cos ${{3\pi } \over 5}$
<br><br> = 2(1 - sin18)
<br><br>= 2(1 - ${{\sqrt 5 - 1} \over 4}$)
<br><br>= 2$\left( {{{5 - \sqrt 5 } \over 4}} \right)$
<br><br>= ${{{5 - \sqrt 5 } \over 2}}$ $\simeq$ 1.385
<br><br>$\therefore$ detB $\in$ (1, 2)
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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