If A=[0-tan α2tan α20] and I is the identity matrix of order 2, show that I + A = (I-A)[cosα-sinαsinαcosα] - Mathematics

प्रश्न

if $A = [\begin{matrix} 0 & - \tan \frac{α}{2} \\ \tan \frac{α}{2} & 0 \end{matrix}]$ and I is the identity matrix of order 2, show that I + A = $(I - A) [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]$

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उत्तर

$A = [\begin{matrix} 0 & - \tan \frac{α}{2} \\ \tan \frac{α}{2} & 0 \end{matrix}], I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

$I + A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] + [\begin{matrix} 0 & - \tan \frac{α}{2} \\ \tan \frac{α}{2} & 0 \end{matrix}]$

$= [\begin{matrix} 1 & - \tan \frac{α}{2} \\ \tan \frac{α}{2} & 1 \end{matrix}]$

$(I - A) [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}] = ([\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] - [\begin{matrix} 0 & - \frac{\tan α}{2} \\ \frac{\tan α}{2} & 0 \end{matrix}]) [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]$

$= [\begin{matrix} 1 & \tan \frac{α}{2} \\ - \tan \frac{α}{2} & 1 \end{matrix}] [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]$

$= [\begin{matrix} 1 & \tan \frac{α}{2} \\ - \tan \frac{α}{2} & 1 \end{matrix}]$ $[\begin{matrix} \frac{1 - \tan^{2} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \frac{- 2 \tan \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \\ \frac{- 2 \tan \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \frac{1 - \tan^{2} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \end{matrix}]$

$= [\begin{matrix} \frac{1 + \tan^{2} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \frac{- \tan \frac{α}{2} - \tan^{3} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \\ \frac{\tan \frac{α}{2} + \tan^{3} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \frac{1 + \tan^{2} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \end{matrix}]$

$= [\begin{matrix} 1 & - \frac{\tan α}{2} \\ \frac{\tan α}{2} & 1 \end{matrix}]$

Hence, $I + A = (I - A) [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]$

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Types of Matrices

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Q 18 Q 17 Q 19.1

पाठ 3: Matrices - Exercise 3.2 [पृष्ठ ८२]

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पाठ 3 Matrices
Exercise 3.2 | Q 18 | पृष्ठ ८२

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If A=[0-tan α2tan α20] and I is the identity matrix of order 2, show that I + A = (I-A)[cosα-sinαsinαcosα] - Mathematics

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If A=[0-tan α2tan α20] and I is the identity matrix of order 2, show that I + A = (I-A)[cosα-sinαsinαcosα] - Mathematics

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