A + A' = I के लिए α का मान क्या है?

Steps to Solve

1. Understanding the Matrix A and its Transpose Given matrix ( A ) is defined as: $$ A = \begin{bmatrix} \cos \alpha & -\sin \alpha \ \sin \alpha & \cos \alpha \end{bmatrix} $$ The transpose of matrix ( A ), denoted as ( A' ), is: $$ A' = \begin{bmatrix} \cos \alpha & \sin \alpha \ -\sin \alpha & \cos \alpha \end{bmatrix} $$

2. Calculating A + A' Now we will add ( A ) and ( A' ): $$ A + A' = \begin{bmatrix} \cos \alpha & -\sin \alpha \ \sin \alpha & \cos \alpha \end{bmatrix} + \begin{bmatrix} \cos \alpha & \sin \alpha \ -\sin \alpha & \cos \alpha \end{bmatrix} $$ Performing the addition results in: $$ A + A' = \begin{bmatrix} 2\cos \alpha & 0 \ 0 & 2\cos \alpha \end{bmatrix} = 2\cos \alpha \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} $$

3. Setting A + A' Equal to the Identity Matrix According to the problem, we need: $$ A + A' = I $$ where ( I ) is the identity matrix: $$ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} $$ Thus, we equate: $$ 2\cos \alpha = 1 $$

4. Solving for α From the equation ( 2\cos \alpha = 1 ), we isolate ( \cos \alpha ): $$ \cos \alpha = \frac{1}{2} $$ The general solutions for ( \alpha ) are: $$ \alpha = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad \alpha = -\frac{\pi}{3} + 2k\pi, \quad k \in \mathbb{Z} $$