Eigenvalues and Eigenvectors MCQ

1. Consider the matrix \begin{bmatrix} 5 & -1\\ 4 & 1 \end{bmatrix}. 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.
Answer
Answer. a

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
Answer
Answer. a

3. Consider the 5 x 5 matrix

    \[A=\begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 5 & 1 & 2 & 3 & 4\\ 4 & 5 & 1 & 2 & 3\\ 3 & 4 & 5 & 1 & 2\\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}\]

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
Answer
Answer. c

4. The matrix A=\begin{bmatrix} \frac{3}{2} & 0 & \frac{1}{2}\\ 0 & -1 & 0\\ \frac{1}{2} & 0 & \frac{3}{2} \end{bmatrix} has three distinct eigen values and one of its eigen vectors is \begin{bmatrix} 1\\ 0\\ 1 \end{bmatrix}. Which one of the following can be another eigen vector of A?

  1. \begin{bmatrix} 0\\ 0\\ -1 \end{bmatrix}
  2. \begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix}
  3. \begin{bmatrix} 1\\ 0\\ -1 \end{bmatrix}
  4. \begin{bmatrix} 1\\ -1\\ 1 \end{bmatrix}
Answer
Answer. c

5. The eigen values of the matrix given below are

    \[\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & -3 & -4 \end{bmatrix}\]

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

6. The eigen values of the matrix A=\begin{bmatrix} 1 & -1 & 5\\ 0 & 5 & 6\\ 0 & -6 & 5 \end{bmatrix} are

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

7. Consider the matrix P=\begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 1 & 0\\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{bmatrix}.

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
Answer
Answer. d

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

P =\begin{bmatrix} 2 & 0 & 1\\ 4 & -3 & 3\\ 0 & 2 & -1 \end{bmatrix}

  1. -6
  2. 2
  3. 6
  4. -2
Answer
Answer. b

9. Consider the matrix A=\begin{bmatrix} 50 & 70\\ 70 & 80 \end{bmatrix} whose eigen vectors corresponding to eigen values λ1 and λ2 are x_1=\begin{bmatrix} 70\\ \lambda_1-50 \end{bmatrix}\; and\; x_2=\begin{bmatrix} \lambda_2-80\\ 70 \end{bmatrix} respectively. The value of x1Tx2 is

  1. 0
  2. 1
  3. 2
  4. 4
Answer
Answer. a

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
Answer
Answer. c

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
Answer
Answer. d

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
Answer
Answer. a

13. Consider the matrix A=\begin{bmatrix} 2 & 1 & 1\\ 2 & 3 & 4\\ -1 & -1 & -2 \end{bmatrix} whose eigen values are 1, -1 and 3. Then trace of (A3 – 3A2) is

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

14. The value of x for which the matrix A=\begin{bmatrix} 3 & 2 & 4\\ 9 & 7 & 13\\ -6 & -4 & -9+x \end{bmatrix} has zero as an eigen value is

  1. 2
  2. 1
  3. 3
  4. 4
Answer
Answer. b

15. The number of linearly independent eigen vecctors of matrix A=\begin{bmatrix} 2 & 1 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{bmatrix} is

  1. 2
  2. 1
  3. 3
  4. 4
Answer
Answer. a
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