Eigenvalues and Eigenvectors MCQ

31. Consider the following simultaneous equations (with c₁ and c₂ being constants):

3x₁ + 2x₂ = c₁
4x₁ + x₂ = c₂

The characterisitics equation for these simultaneous equations is

1. λ² – 4λ – 5 = 0
2. λ² – 4λ + 5 = 0
3. λ² + 4λ – 5 = 0
4. λ² + 4λ + 5 = 0

32. All the four entries of the 2 x 2 matrix $\begin{bmatrix} p_{11} & p_{12}\\ p_{21} & p_{22} \end{bmatrix}$ are nonzero, and one of its eigen values is zero. Which of the following statements is true?

1. p₁₁p₂₂ – p₁₂p₂₁ = 1
2. p₁₁p₂₂ – p₁₂p₂₁ = -1
3. p₁₁p₂₂ – p₁₂p₂₁ = 0
4. p₁₁p₂₂ + p₁₂p₂₁ = 0

33. If a square matrix A is real and symmetric, then the eigen values

1. are always real
2. are always real and positive
3. are always real and non-negative
4. occur in complex conjugate pairs

34. Which one of the following statements is true about every n x n matrix with only real eigen values?

1. if the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative.
2. if the trace of the matrix is positive, all its eigen values are positive.
3. if the determinant of the matrix is positive, all its eigen values are positive.
4. if the product of the trace and determinant of the matrix is positive, all its eigen values are positive.

35. The product of the non-zero eigen values of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1\\ 0 & 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is

1. 2
2. 3
3. 6
4. 8

36. The value of the dot product of the eigen vectors corresponding to any pair of different eigen values of a 4 x 4 symmetric positive definite matrix is

1. -2
2. 0
3. 6
4. 10

37. A real (4 x 4) matrix A satisfies the equation A² = I, where I is the (4 x 4) identity matrix. The positive eigen value of A is

1. +2
2. +3
3. +1
4. +6

38. A system matrix is given as follows.

\[A=\begin{bmatrix} 0 & 1 & -1\\ -6 & -11 & 6\\ -6 & -11 & 5 \end{bmatrix}\]

The absolute value of the ratio of the maximum eigen value to the minimum eigen value is

1. 2
2. 3
3. 6
4. 8

39. One of the eigen vectors of matrix $\begin{bmatrix} -5 & 2\\ -9 & 6 \end{bmatrix}$

1. \[\begin{Bmatrix} -1\\ 1 \end{Bmatrix}\]
2. \[\begin{Bmatrix} -2\\ 9 \end{Bmatrix}\]
3. \[\begin{Bmatrix} 2\\ -1 \end{Bmatrix}\]
4. \[\begin{Bmatrix} 1\\ 1 \end{Bmatrix}\]

40. Consider a 3 x 3 real symmetric matrix S such that two of its eigen values are a≠0, b≠0 with respective eigen vectors $\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}\: , \begin{bmatrix} y_1\\ y_2\\ y_3 \end{bmatrix}$. If a ≠ b then x₁y₁ + x₂y₂ + x₃y₃ equals

1. a
2. b
3. ab
4. 0

41. The sum of eigen values of matrix, [M] is

\[[M]=\begin{bmatrix} 215 & 650 & 795\\ 655 & 150 & 835\\ 485 & 355 & 550 \end{bmatrix}\]

1. 915
2. 1355
3. 1640
4. 2180

42. Which one of the following statements is not true for a square matrix A?

1. if A is upper triangular matrix, the eigen values of A are the diagonal elements of it
2. if A is real symmetric matrix, the eigen values of A are always real and positive
3. if A is real, the eigen values of A and A^T are always the same
4. if all the principal minors of A are positive, all the eigen values of A are also positive

43. Which one of the following statements is true for all real symmetric matrices?

1. all the eigen values are real
2. all the eigen values are positive
3. all the eigen values are distinct
4. sum of all the eigen values is zero

44. The minimum eigen value of the following matrix is

\[\begin{bmatrix} 3 & 5 & 2\\ 5 & 12 & 7\\ 2 & 7 & 5 \end{bmatrix}\]

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

45. A matrix has eigen values -1 and -2. The corresponding eigen vectors are $\begin{bmatrix} 1\\ -1 \end{bmatrix}\; and \; \begin{bmatrix} 1\\ -2 \end{bmatrix}$ respectively. The matrix is

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

46. The eigen values of a symmetric matrix are all

1. complex with non-zero positive imaginary part
2. complex with non-zero negative imaginary part
3. real
4. pure imaginary

47. One pair of eigen vectors correponding to the two eigen values of the matrix $\begin{bmatrix} 1 & 1\\ -1 & -2 \end{bmatrix}$ is

1. \[\begin{bmatrix} 1\\ -j \end{bmatrix}\; ,\; \begin{bmatrix} j\\ -1 \end{bmatrix}\]
2. \[\begin{bmatrix} 0\\ 1 \end{bmatrix}\; ,\; \begin{bmatrix} -1\\ 0 \end{bmatrix}\]
3. \[\begin{bmatrix} 1\\ j \end{bmatrix}\; ,\; \begin{bmatrix} 0\\ 1 \end{bmatrix}\]
4. \[\begin{bmatrix} 1\\ j \end{bmatrix}\; ,\; \begin{bmatrix} j\\ 1 \end{bmatrix}\]

48. The eigen values of matrix $\begin{bmatrix} 9 & 5\\ 5 & 8 \end{bmatrix}$ are

1. -2.42 and 6.86
2. 3.48 and 13.53
3. 4.70 and 6.86
4. 6.86 and 9.50

49. Consider the matrix as given below:

\[\begin{bmatrix} 1 & 2 & 3\\ 0 & 4 & 7\\ 0 & 0 & 3 \end{bmatrix}\]

Which one of the following options provides the correct values of the eigen values of the matrix?

1. 1, 4, 3
2. 3, 7, 3
3. 7, 3, 2
4. 1, 2, 3

50. Eigen values of a real symmetric matrix are always

1. positive
2. negative
3. real
4. complex

51. Consider the following matrix

\[A=\begin{bmatrix} 2 & 3\\ x & y \end{bmatrix}\]

If the eigen values of A are 4 and 8, then

1. x = 4, y = 10
2. x = 5, y = 8
3. x = -3, y = 9
4. x = -4, y = 10

52. The eigen values of a skew-symmetric matrix are

1. always zero
2. always pure imaginary
3. either zero or pure imaginary
4. always real

53. An eigen vector of $P=\begin{bmatrix} 1 & 1 & 0\\ 0 & 2 & 2\\ 0 & 0 & 3 \end{bmatrix}$ is

1. [-1  1  1]^T
2. [1  2  1]^T
3. [1  -1  2]^T
4. [2  1  -1]^T

54. One of the eigen vectors of the matrix $A=\begin{bmatrix} 2 & 2\\ 1 & 3 \end{bmatrix}$

1. \[\begin{Bmatrix} 2\\ -1 \end{Bmatrix}\]
2. \[\begin{Bmatrix} 2\\ 1 \end{Bmatrix}\]
3. \[\begin{Bmatrix} 4\\ 1 \end{Bmatrix}\]
4. \[\begin{Bmatrix} 1\\ -1 \end{Bmatrix}\]

55. The eigen values of the following matrix are

\[P=\begin{bmatrix} -1 & 3 & 5\\ -3 & -1 & 6\\ 0 & 0 & 3 \end{bmatrix}\]

1. 3, 3 + 5j, 6 – j
2. -6 + 5j, 3 + j, 3 – j
3. 3 + j, 3 – j, 5 + j
4. 3, -1 + 3j, -1 – 3j

56. The trace and determinant of a 2 x 2 matrix are known to be -2 and -35 respectively. It eigen values are

1. -30 and -5
2. -37 and -1
3. -7 and 5
4. 17.5 and -2

57. The matrix $\begin{bmatrix} 1 & 2 & 4\\ 3 & 0 & 6\\ 1 & 1 & p \end{bmatrix}$ has one eigen value equal to 3. The sum of the other two eigen values is

1. p
2. p-1
3. p-2
4. p-3

58. How many of the following matrices have an eigen value 1?

\[\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix}and \begin{bmatrix} -1 & 0\\ 1 & -1 \end{bmatrix},\]

1. one
2. two
3. three
4. four

59. The eigen vectors of the matrix $\begin{bmatrix} 1 & 2\\ 0 & 2 \end{bmatrix}$ are written in the form $\begin{bmatrix} 1\\ a \end{bmatrix} and \begin{bmatrix} 1\\ b \end{bmatrix}$. What is a+b?

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

60. The eigen values of the matrix $P=\begin{bmatrix} 4 & 5\\ 2 & -5 \end{bmatrix}$ are

1. -7 and 8
2. -6 and 5
3. 3 and 4
4. 1 and 2