Two eigenvalues of a 3 x 3 real matrix P are (2 + sqrt(-1)) and 3. The determinant of P is _______.

GATE 2016 · Engineering Mathematics · Eigen Value · medium

Answer: 15

1. Identify the conjugate pair: Given lambda_1 = 2+i, the conjugate lambda_2 = 2-i must also be an eigenvalue of P.
2. Compute the product of the conjugate pair: lambda_1 * lambda_2 = (2+i)(2-i) = 2^2 + 1^2 = 4 + 1 = 5.
3. Compute determinant as product of all eigenvalues: det(P) = lambda_1 * lambda_2 * lambda_3 = 5 * 3 = 15.