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 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
- 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 λ1 and λ2 are
respectively. The value of x1Tx2 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, P3 = 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 (A3 – 3A2) 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