Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, and 4. Which of the…
2007
Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, and 4. Which of the following is an eigenvalue of the block matrix
[A I]
[I A]
where I is the 4 × 4 identity matrix?
- A.
-5
- B.
-7
- C.
2
- D.
1
Attempted by 3 students.
Show answer & explanation
Correct answer: C
Let v be an eigenvector of A corresponding to eigenvalue λ, so Av = λv.
For the block matrix B = [A I; I A]:
B [v; v] = [Av + v; v + Av] = [(λ + 1)v; (λ + 1)v], so λ + 1 is an eigenvalue.
B [v; -v] = [Av - v; v - Av] = [(λ - 1)v; -(λ - 1)v], so λ - 1 is also an eigenvalue.
Therefore, for eigenvalues -5, -2, 1, and 4 of A, the block matrix has eigenvalues:
-6, -4, -3, -1, 0, 2, 3, and 5.
Among the options, 2 is an eigenvalue.