Question

Let \( P \) be a \( 4 \times 4 \) matrix with entries from the set of rational numbers. If \( \sqrt{2} + i \), with \( i = \sqrt{-1} \), is a root of the characteristic polynomial of \( P \) and \( I \) is the \( 4 \times 4 \) identity matrix, then

• A. \( P^4 = 4P^2 + 9I \)
• B. \( P^4 = 4P^2 - 9I \)
• C. \( P^4 = 2P^2 - 9I \)
• D. \( P^4 = P^2 + 9I \)

Hint:

When dealing with matrices with complex eigenvalues, use the characteristic polynomial and its properties to find relations between powers of the matrix.

Answer 1

The Correct Option is C

Step 1: Use the fact that \( \sqrt{2} + i \) is a root of the characteristic polynomial.
Let \( \lambda = \sqrt{2} + i \) be an eigenvalue of \( P \). The characteristic polynomial of \( P \) will have \( \lambda \) as a root, and its conjugate \( \bar{\lambda} = \sqrt{2} - i \) will also be a root due to the rational entries of the matrix \( P \). The characteristic polynomial must therefore be of the form: \[ (\lambda - (\sqrt{2} + i)) (\lambda - (\sqrt{2} - i)). \]
Step 2: Expand the characteristic polynomial.
Expanding this product gives: \[ (\lambda - (\sqrt{2} + i)) (\lambda - (\sqrt{2} - i)) = (\lambda^2 - 2\sqrt{2} \lambda + 9). \]
Step 3: Relate this to \( P^2 \) and \( P^4 \).
Since \( \lambda \) is an eigenvalue of \( P \), we substitute \( P \) for \( \lambda \) in the characteristic polynomial. We now know that \( P^2 - 2\sqrt{2}P + 9I = 0 \). Squaring both sides of the equation, we get: \[ P^4 = 4P^2 + 9I. \]
Step 4: Conclusion.
Thus, the correct answer is \( \boxed{(A)} \).