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.

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

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

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. $\begin{bmatrix} 1\\ -1\\ 1 \end{bmatrix}$

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)

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

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

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

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 x₁^Tx₂ is

1. 0
2. 1
3. 2
4. 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. 1
2. 3
3. 5
4. 25

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

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

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

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

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