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

- eigen value 3 has a multiplicity of 2, and only one independent eigen vector exists.
- eigen value 3 has a multiplicity of 2, and two independent eigen vector exists.
- eigen value 3 has a multiplicity of 2, and no independent eigen vector exists.
- 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 λ^{3} – 4λ^{2} + 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

- 5
- 2
- 3
- 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

- -2.5
- 0
- 15
- 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

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

6. The eigen values of the matrix are

- -1, 5, 6
- 1, -5 ± j6
- 1, 5 ± j6
- 1, 5, 5

7. Consider the matrix .

Which one of the following statements about P is incorrect?

- determinant of P is equal to 1
- P is orthogonal
- inverse of P is equal to its transpose
- all eigen values of P are real numbers

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

- -6
- 2
- 6
- -2

9. Consider the matrix whose eigen vectors corresponding to eigen values λ_{1} and λ_{2} are respectively. The value of x_{1}^{T}x_{2} is

- 0
- 1
- 2
- 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
- 3
- 5
- 25

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

- 1, 1, -1
- 1, 0.5 + j0.866, 0.5 -j0.866
- 1, -0.5 + j0.866, -0.5 – j0.866
- 0, 1, -1

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

- 0.125
- 0.225
- 0.200
- 0.140

13. Consider the matrix whose eigen values are 1, -1 and 3. Then trace of (A^{3} – 3A^{2}) is

- 6
- -6
- 5
- -5

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

- 2
- 1
- 3
- 4

15. The number of linearly independent eigen vecctors of matrix is

- 2
- 1
- 3
- 4