# Eigenvalues and Eigenvectors MCQ

1. Consider the matrix . Which one of the following statements is true for the eigen values and eigen vectors of this matrix?

1. eigen value 3 has a multiplicity of 2, and only one independent eigen vector exists.
2. eigen value 3 has a multiplicity of 2, and two independent eigen vector exists.
3. eigen value 3 has a multiplicity of 2, and no independent eigen vector exists.
4. eigen value are 3 and -3, and two independent eigen vectors exist.

2. If the characteristic polynomial of a 3 x 3 matrix M over R (the set of real numbers) is λ3 – 4λ2 + aλ + 30, a ∈ R and one eigen value of M is 2. Then the largest among the absolute values of the eigen values of M is

1. 5
2. 2
3. 3
4. 6

3. Consider the 5 x 5 matrix

It is given that A has only one real eigen value. Then the real eigen value of A is

1. -2.5
2. 0
3. 15
4. 25

4. The matrix has three distinct eigen values and one of its eigen vectors is . Which one of the following can be another eigen vector of A?

5. The eigen values of the matrix given below are

1. (0, -1, -3)
2. (0, -2, -3)
3. (0, 2, 3)
4. (0, 1, 3)

6. The eigen values of the matrix are

1. -1, 5, 6
2. 1, -5 ± j6
3. 1, 5 ± j6
4. 1, 5, 5

7. Consider the matrix .

Which one of the following statements about P is incorrect?

1. determinant of P is equal to 1
2. P is orthogonal
3. inverse of P is equal to its transpose
4. all eigen values of P are real numbers

8. The product of eigen values of the matrix P is

1. -6
2. 2
3. 6
4. -2

9. Consider the matrix whose eigen vectors corresponding to eigen values λ1 and λ2 are respectively. The value of x1Tx2 is

1. 0
2. 1
3. 2
4. 4

10. The determinant of a 2 x 2 matrix is 50. If one eigen value of the matrix is 10, the other eigen value is

1. 1
2. 3
3. 5
4. 25

11. A 3 x 3 matrix P is such that, P3 = P. Then the eigen values of P are

1. 1, 1, -1
2. 1, 0.5 + j0.866, 0.5 -j0.866
3. 1, -0.5 + j0.866, -0.5 – j0.866
4. 0, 1, -1

12. Suppose that the eigen values of matrix A are 1, 2, 4. The determinant of (A-1)T is

1. 0.125
2. 0.225
3. 0.200
4. 0.140

13. Consider the matrix whose eigen values are 1, -1 and 3. Then trace of (A3 – 3A2) is

1. 6
2. -6
3. 5
4. -5

14. The value of x for which the matrix has zero as an eigen value is

1. 2
2. 1
3. 3
4. 4