Consider the matrix M = [[2, -1], [3, 1]]. Which ONE of the following statements is TRUE?
A. The eigenvalues of M are no...

Question

Consider the matrix M = [[2, -1], [3, 1]]. Which ONE of the following statements is TRUE?
A. The eigenvalues of M are non-negative and real.
B. The eigenvalues of M are complex conjugate pairs.
C. One eigenvalue of M is positive and real, and another eigenvalue of M is zero.
D. One eigenvalue of M is non-negative and real, and another eigenvalue is negative and real.

Accepted Answer

Eigenvalues are (3/2 + i*sqrt(11)/2) and (3/2 - i*sqrt(11)/2): complex conjugate pair. Answer: B. For a 2x2 matrix, eigenvalues are found from the characteristic polynomial det(M - lambda*I) = 0. If the discriminant of this quadratic is negative, the eigenvalues are complex conjugates. The matrix M = [[2,-1],[3,1]] has trace = 3 and determinant = 2*1 - (-1)*3 = 5, giving characteristic polynomial lambda^2 - 3*lambda + 5 = 0 with discriminant 9 - 20 = -11 < 0.