Question

Let \( Y \) follow \( N_8(0, I_8) \) distribution, where \( I_8 \) is the \( 8 \times 8 \) identity matrix. Let \( Y^T \Sigma_1 Y \) and \( Y^T \Sigma_2 Y \) be independent and follow central chi-square distributions with 3 and 4 degrees of freedom, respectively, where \( \Sigma_1 \) and \( \Sigma_2 \) are \( 8 \times 8 \) matrices and \( Y^T \) denotes transpose of \( Y \). Then which of the following statements is/are true? \[ P: \Sigma_1 \, \text{and} \, \Sigma_2 \, \text{are idempotent.} \quad Q: \Sigma_1 \Sigma_2 = 0, \, \text{where} \, 0 \, \text{is the} \, 8 \times 8 \, \text{zero matrix.} \]

• A. P only
• B. Q only
• C. Both P and Q
• D. Neither P nor Q

Hint:

In matrix theory, the idempotent property means that a matrix multiplied by itself equals the matrix. For orthogonal matrices, their product is zero, implying that they are independent or orthogonal.

Answer 1

The Correct Option is C

We are given a multivariate normal distribution \( Y \) with specific conditions on its transformation. We are asked to determine the truth of two statements: \( P \) and \( Q \).
Step 1: Understanding the idempotent property.
A matrix \( \Sigma \) is idempotent if \( \Sigma^2 = \Sigma \). Given the structure of the problem, we know that both \( \Sigma_1 \) and \( \Sigma_2 \) are matrices associated with chi-square distributed variables, and they have the idempotent property. This means that both \( P \) and \( Q \) are true.
Step 2: Analyzing the zero matrix condition.
The condition \( \Sigma_1 \Sigma_2 = 0 \) implies that \( \Sigma_1 \) and \( \Sigma_2 \) are orthogonal matrices. This condition holds in this case, confirming that both \( P \) and \( Q \) are correct.
Step 3: Conclusion.
The correct answer is (C), as both statements \( P \) and \( Q \) are true.
Final Answer: \[ \boxed{(C) \, \text{Both P and Q are true.}} \]