Question

If \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \) and \( A + A' = I \), then find the value of \( \alpha \).

Hint:

Remember the basic properties of matrices: the transpose A' is found by flipping the matrix over its main diagonal, and matrix addition involves adding corresponding elements. For matrix equality, every corresponding element must be equal.

Answer 1

Step 1: Understanding the Concept:
We are given a matrix A and a condition involving the matrix, its transpose (A'), and the identity matrix (I). We need to solve the resulting matrix equation to find the value of the angle \( \alpha \).
Step 2: Key Formula or Approach:
1. Find the transpose of A, denoted by A', by interchanging its rows and columns.
2. Add the matrices A and A'.
3. Set the resulting matrix equal to the 2x2 identity matrix, \( I = \begin{bmatrix} 1 & 0
0 & 1 \end{bmatrix} \).
4. Equate the corresponding elements to form an equation and solve for \( \alpha \).
Step 3: Detailed Explanation or Calculation:
The given matrix is:
\[ A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \] Its transpose is:
\[ A' = \begin{bmatrix} \cos\alpha & \sin\alpha
-\sin\alpha & \cos\alpha \end{bmatrix} \] Now, add A and A':
\[ A + A' = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} + \begin{bmatrix} \cos\alpha & \sin\alpha
-\sin\alpha & \cos\alpha \end{bmatrix} \] \[ A + A' = \begin{bmatrix} \cos\alpha + \cos\alpha & -\sin\alpha + \sin\alpha
\sin\alpha - \sin\alpha & \cos\alpha + \cos\alpha \end{bmatrix} = \begin{bmatrix} 2\cos\alpha & 0
0 & 2\cos\alpha \end{bmatrix} \] We are given that \( A + A' = I \):
\[ \begin{bmatrix} 2\cos\alpha & 0
0 & 2\cos\alpha \end{bmatrix} = \begin{bmatrix} 1 & 0
0 & 1 \end{bmatrix} \] Equating the corresponding elements (specifically, the diagonal elements):
\[ 2\cos\alpha = 1 \] \[ \cos\alpha = \frac{1}{2} \] The principal value of \( \alpha \) for which \( \cos\alpha = \frac{1}{2} \) is \( \alpha = \frac{\pi}{3} \).
Step 4: Final Answer:
The value of \( \alpha \) is \( \frac{\pi}{3} \).