Eigenvalues and Eigenvectors MCQ

16. Let the eigen values of a 2 x 2 matrix A be 1, -2 with eigen vectors x₁ and x₂ respectively. Then the eigen values and eigen vectors of the matrix A² – 3A + 4I would, respectively, be

1. 2, 14; x₁, x₂
2. 2, 14; x₁+ x₂, x₁– x₂
3. 2, 0; x₁, x₂
4. 2, 0; x₁+ x₂, x₁– x₂

17. Two eigen values of a 3 x 3 real matrix P are $2+\sqrt{-1}$ and 3. The determinant of P is

1. 5
2. 10
3. 15
4. 20

18. Consider a 2 x 2 square matrix

\[A=\begin{bmatrix} \sigma & x\\ \omega & \sigma \end{bmatrix}\]

where x is unknown. If the eigen values of the matrix A are (σ + jω) and (σ – jω), then x is equal to

1. + jω
2. – jω
3. + ω
4. – ω

19. The condition for which the eigen values of the matrix $A=\begin{bmatrix} 2 & 1\\ 1 & k \end{bmatrix}$ are positive, is

1. k > 1/2
2. k > -2
3. k > 0
4. k < -1/2

20. Consider a 3 x 3 matrix with every element being equal to 1. Its only non-zero eigen value is

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

21. If the entries in each column of a square matrix M add up to 1, then an eigen value of M is

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

22. Consider the following 2 x 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are -1 and 7. What are the values of a and b?

$A=\begin{bmatrix} 1 & 4\\ b & a \end{bmatrix}$

1. a = 6, b = 4
2. a = 4, b = 6
3. a = 3, b = 5
4. a = 5, b = 3

23. The value of x for which all the eigen-values of the matrix given below are real is

\[\begin{bmatrix} 10 & 5+j & 4\\ x & 20 & 2\\ 4 & 2 & -10 \end{bmatrix}\]

1. 5 + j
2. 5 – j
3. 1 – 5j
4. 1 + 5j

24. At least one eigen value of a singular matrix is

1. positive
2. zero
3. negative
4. imaginary

25. The two eigen values of the matrix $\begin{bmatrix} 2 & 1\\ 1 & p \end{bmatrix}$ have a ratio of 3 : 1 for p = 2. What is another value of p for which the eigen values have the same ratio of 3 : 1?

1. -2
2. 1
3. 7/3
4. 14/3

26. The larger of the two eigen values of the matrix $\begin{bmatrix} 4 & 5\\ 2 & 1 \end{bmatrix}$ is

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

27. The value of p such that the vector $\begin{bmatrix} 1\\ 2 \\ 3 \end{bmatrix}$ is an eigen vector of the matrix $\begin{bmatrix} 4 & 1 & 2\\ p & 2 & 1\\ 14 & -4 & 10 \end{bmatrix}$ is

1. 17
2. 16
3. 15
4. 14

28. The lowest eigen value of the 2 x 2 matrix $\begin{bmatrix} 4 & 2\\ 1 & 3 \end{bmatrix}$ is

1. 5
2. 2
3. 3
4. 4

29. The smallest and largest eigen values of the following matrix are

$\begin{bmatrix} 3 & -2 & 2 \\ 4 & -4 & 6 \\ 2 & -3 & 5 \end{bmatrix}$

1. 1.5 and 2.5
2. 0.5 and 2.5
3. 1 and 3
4. 1 and 2

30. In the given matrix $\begin{bmatrix} 1 & -1 & 2\\ 0 & 1 & 0\\ 1 & 2 & 1 \end{bmatrix}$, one of the eigen values is 1. The eigen vectors corresponding to the eigen value 1 are

1. {α(4,2,1) | α ≠ 0, α ∈ R}
2. {α(-4,2,1) | α ≠ 0, α ∈ R}
3. {α($\sqrt{2}$,0,1) | α ≠ 0, α ∈ R}
4. {α($-\sqrt{2}$,0,1) | α ≠ 0, α ∈ R}