Consider the matrix \(M = \begin{bmatrix} 2 & -1 \\ 3 & 1 \\ \end{bmatrix} \).…
2024
Consider the matrix \(M = \begin{bmatrix} 2 & -1 \\ 3 & 1 \\ \end{bmatrix} \).
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 𝑴is positive and real, and another eigenvalue of 𝑴 is zero.
- D.
One eigenvalue of 𝑴 is non-negative and real, and another eigenvalue of 𝑴 is negative and real.
Attempted by 5 students.
Show answer & explanation
Correct answer: B
Find the characteristic polynomial using trace and determinant.
Trace = 2 + 1 = 3, Determinant = 2·1 - (-1)·3 = 2 + 3 = 5.
Characteristic polynomial: λ^2 - (trace)λ + determinant = λ^2 - 3λ + 5.
Discriminant Δ = 3^2 - 4·1·5 = 9 - 20 = -11 < 0, so the roots are not real but form a complex conjugate pair.
Explicit eigenvalues: (3 ± i√11)/2.
Conclusion: the statement that the eigenvalues are complex conjugate pairs is true. Statements asserting real eigenvalues, a zero eigenvalue, or one positive and one negative real eigenvalue are all false because the discriminant is negative (so eigenvalues are nonreal) and the determinant is 5 (so no eigenvalue is zero and the product is positive).