Eigenvalues and Eigenvectors MCQ

61. The minimum and the maximum eigen values of the matrix $\begin{bmatrix} 1 & 1 & 3\\ 1 & 5 & 1\\ 3 & 1 & 1 \end{bmatrix}$ are -2 and 6, respectively. What is the other eigen value?

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

62. The number of linearly independent eigen vectors of $\begin{bmatrix} 2 & 1\\ 0 & 2 \end{bmatrix}$ is

  1. 0
  2. 1
  3. 2
  4. infinite
Answer
Answer. b

63. Eigen values of a matrix $\begin{bmatrix} 3 & 2\\ 2 & 3 \end{bmatrix}$ are 5 and 1. What are the eigen values of the matrix S2 = SS?

  1. 1 and 25
  2. 6 and 4
  3. 5 and 1
  4. 2 and 10
Answer
Answer. a

64. For a given matrix $\begin{bmatrix} 2 & -2 & 3\\ -2 & -1 & 6\\ 1 & 2 & 0 \end{bmatrix}$, one of the eigen values is 3. The other two eigen values are

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

65. For the matrix $\begin{bmatrix} 4 & 2\\ 2 & 4 \end{bmatrix}$ the eigen value corresponding to the eigen vector $\begin{bmatrix} 101\\ 101 \end{bmatrix}$ is

  1. 2
  2. 4
  3. 6
  4. 8
Answer
Answer. c

66. What are the eigen values of the following 2 x 2 matrix?

$\begin{bmatrix} 2 & -1\\ -4 & 5 \end{bmatrix}$

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

67. Given the matrix $\begin{bmatrix} -4 & 2\\ 4 & 3 \end{bmatrix}$

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

67. For the matrix $\begin{bmatrix} 3 & -2 & 2\\ 0 & -2 & 1\\ 0 & 0 & 1 \end{bmatrix}$, one of the eigen values is equal to -2. Which of the following is an eigen vector?

  1. \[\begin{bmatrix} 3\\ -2\\ 1 \end{bmatrix}\]
  2. \[\begin{bmatrix} -3\\ 2\\ -1 \end{bmatrix}\]
  3. \[\begin{bmatrix} 1\\ -2\\ 3 \end{bmatrix}\]
  4. \[\begin{bmatrix} 2\\ 5\\ 0 \end{bmatrix}\]
Answer
Answer. d

68. Which one of the following is an eigen vector of the matrix $\begin{bmatrix} 5 & 0 & 0 & 0\\ 0 & 5 & 5 & 0\\ 0 & 0 & 2 & 1\\ 0 & 0 & 3 & 1 \end{bmatrix}$?

  1. \[\begin{bmatrix} 1\\ -2\\ 0 \\ 0 \end{bmatrix}\]
  2. \[\begin{bmatrix} 0\\ 0\\ 1 \\ 0 \end{bmatrix}\]
  3. \[\begin{bmatrix} 1\\ 0\\ 0 \\ -2 \end{bmatrix}\]
  4. \[\begin{bmatrix} 1\\ -1\\ 2 \\ 1 \end{bmatrix}\]
Answer
Answer. a

69. The sum of the eigen values of the matrix given below is $\begin{bmatrix} 1 & 2 & 3\\ 1 & 5 & 1\\ 3 & 1 & 1 \end{bmatrix}$.

  1. 5
  2. 7
  3. 9
  4. 18
Answer
Answer. b

70. The eigen values of the matrix $\begin{bmatrix} 4 & -2\\ -2 & 1 \end{bmatrix}$

  1. are 1 and 4
  2. are -1 and 2
  3. are 0 and 5
  4. cannot be determined
Answer
Answer. c

71. For the matrix $\begin{bmatrix} 4 & 1\\ 1 & 4 \end{bmatrix}$ the eigen values are

  1. 3 and -3
  2. -3 and -5
  3. 3 and 5
  4. 5 and 0
Answer
Answer. c
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