What are the eigenvalues of the matrix A = [[2, -1], [-4, 5]]?
2005
What are the eigenvalues of the matrix A = [[2, -1], [-4, 5]]?
- A.
-1 and 1
- B.
1 and 6
- C.
2 and 5
- D.
4 and -1
Attempted by 4 students.
Show answer & explanation
Correct answer: B
For A = [[2, -1], [-4, 5]], compute the characteristic equation det(A - λI) = 0.
A - λI = [[2 - λ, -1], [-4, 5 - λ]].
So,
det(A - λI) = (2 - λ)(5 - λ) - (-1)(-4)
= (10 - 7λ + λ^2) - 4
= λ^2 - 7λ + 6.
Now factor:
λ^2 - 7λ + 6 = (λ - 1)(λ - 6).
Therefore the eigenvalues are 1 and 6.