Step 1: Check transitivity. If \( (a, b) \in R \) and \( (b, c) \in R \), then \( a = b^2 \) and \( b = c^2 \).
Step 2: Check if \( (a, c) \in R \). \[ a = (c^2)^2 = c^4 \] Since \( a \neq c^2 \), transitivity does not hold. Example: Take \( a = 16, b = 4, c = 2 \). \[ (16,4) \in R \text{ and } (4,2) \in R \] But \( (16,2) \notin R \), hence not transitive.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where
\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]