MTH603-Numerical Analysis Quiz MCQS #Objective #Questions #MidTerm

1. If n x n matrices A and B are similar, then they have the ___ eigenvalues (with the same multiplicities)

2. If the product of two matrices is an identity matrices that is AB = I, then which of the following is true?

- A is transpose of B
- A is inverse of B ✔
- A is singular
- B is singular

3. Iterative algorithms can be more rapid than direct methods.

4. While using Relaxation method, which of the following is the largest Residual for 1st iteration on the system; 2x + 3y = 1, 3x + 2y = -4?

5. Which of the following systems of linear equations has a strictly diagonally dominant coefficient matrix?

- -2x+7x+2x=5, 6x-2x+3x=1, x+x-5x=-13
- -2x+7x+2x=5, 6x-2x+3x=1, x+x-5x=-13
- x+x-5x=-13, 6x-2x+3x=1, -2x+7x+2x=5
- 6x-2x+3x=1, -2x+7x+2x=5, x+x-5x=-13

6. In the context of Jacobi's method for finding Eigen values and Eigen vectors of a real symmetric matrix of order 2*2, if |-5| be its largest off-diagonal then which of the following will be its corresponding off-diagonal values of Orthogonal Matrix?

- Cos(theta), -Cos(theta)
- Sin(theta), Cos(theta)
- Sin(theta), -Sin(theta)
- -Sin(theta), Cos(theta)

7. While using power method, from the resultant normalize vector

8. Every non-zero vector x is an eigenvector of the identity matrix with Eigen value___

9. Which of the following systems of linear equations has a strictly diagonally coefficient matrix?

- -x+12x+5x=8, 9x+5x-3x=12, 2x-4x+7x=-15
- 9x+5x-3x=12, -x+12x+5x=8, 2x-4x+7x=-15
- 2x-4x+7x=-15, -x+12x+5x=8, 9x+5x-3x=12
- 9x+5x-3x=12, 2x-4x+7x=-15, -x+12x+5x=8

10. Let[A] be a 3 x 3 real symmetric matrix with |a| be numerically the largest off-diagonal element of A, then we can construct orthogonal matrix S1 by Jacobi's method as

11. If the pivot element happens to be zero, then the i-th column elements are searched for the numerically ___ element

12. Exact solution of 2/3 is not exists

13. A and its transpose matrix have ___ eigenvalues

14. If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities)

15. Power method is applicable if the eigen values are real and distinct

16. By using determinants, we can easily check that the solution of the given system of linear equation ___ and it is ___

- exists, unique ✔
- exists, consistent
- trivial, unique
- nontrivial, inconsistent

17. Power method is applicable if the eigen vectors corresponding to eigen values are linearly ___

18. When the condition of diagonal dominance becomes true in Jacobi's Method. Then its means that the method is ___

- Stable
- Unstable
- Convergent ✔
- Divergent

19. While using Jacobi method for the matrix

A = [ | 1 | 1/4 | 1/3 | ] |

1/4 | 1/3 | 1/2 |

1/3 | 1/2 | 1/5 |

the value of 'theta Θ' can be found as

- tan 2Θ = 2a
_{13}/a_{11}-a_{33} ✔

20. While using Jacobi method for the matrix

A = [ | 1 | 1/4 | 1/2 | ] |

1/4 | 1/3 | 1/4 |

1/2 | 1/4 | 1/5 |

and 'theta Θ=0.4480' the orthogonal matrix S1 will be given by

S_{1} = [ | cos 0.4480 | 0 | -sin 0.4480 | ] ✔ |

0 | 1 | 0 |

sin 0.4480 | 0 | cos 0.4480 |

21. Full pivoting, in fact, is more ___ than the partial pivoting

22. While using the Gauss-Seidel Method for finding the solution of the system of equation, the following system

x + 2y + 2z = 3

x + 3y + 3z = 2

x + y + 5z = 2

- x = 3 - 2y - 2z, y = 2/3 - x/3 - z, z = 2/5 - x/5 - y/5

23. By using determinants, we can easily check that the solution of the given system of linear equation exists and it is unique

24. While using Jacobi method for the matrix

A = [ | 1 | 1/4 | 1/3 | ] |

1/4 | 1/3 | 1/2 |

1/3 | 1/2 | 1/5 |

and 'theta Θ= 0.7191' the orthogonal matrix S1 will be given by

S_{1} = [ | cos 0.7191 | 0 | -sin 0.7191 | ] ✔ |

0 | 1 | 0 |

sin 0.7191 | 0 | cos 0.7191 |

25. The linear equation x + y = 1 has ___ solution/solutions

- no solution
- unique ✔
- infinite many
- finite many

26. For a system of linear equations, the corresponding coefficient matrix has the value of determinant; |A|=-3, then which of the following is true?

- The system has unique solution ✔
- The system has finite multiple solutions
- The system has infinite many solutions
- The system has no solution

27. In Gauss-Jacobi's method, the corresponding elements of x_{i}^{(r+1)} replaces those of x_{i}^{r} as soon as they become available

28. An augmented matrix may also be used to find the inverse of a matrix by combining it with the ___ matrix

- Inverse
- Square
- Identity ✔
- None

29. Power method is applicable it the eigen vectors corresponding to eigen values are linearly independent

30. While using the relaxation method for finding the solution of the below given system, which of the following increment will be introduced?

6x_{1} - 2x_{2} + 3x_{3} = 1

-2x_{1} + 7x_{2} + 2x_{3} = 5

x_{1} + x_{2} - 5x_{3} = -13

31. Let |A| be a 3 x 3 real symmetric matrix with |a_{23}| be the numerically largest off-diagonal element then using Jacobi's method the value of theta can be found by

- tan 2 Θ = 2a
_{23}/a_{22}-a_{33 }✔

32. The linear equation; 0x+0y=2 has ___ solution/solutions

- unique
- no solution ✔
- infinite many
- finite many

33. The root of the equation xe^{x}-5=0 is bounded in the interval

- [-2, 1]
- [-1, 1]
- [0, 1] ✔
- [1, 2]

34. Which of the following is a forward difference table for the given values of x and y?

x 0.1 0.5 0.9

y 0.003 0.148 0.370

x | y | Δy | Δ^{2}y |

0.1 | 0.003 | 0.145 | 0.077 |

0.5 | 0.148 | 0.222 | | ✔ |

0.9 | 0.37 | | |

35. In ___ method, a system is reduced to an equivalent diagonal form using elementary transformations

- Jacobi's
- Gauss-Seidel
- Relaxation
- Gaussian elimination ✔

36. If the determinant of a matrix A is not equal to zero then the system of equations will have ___

- a unique solution ✔
- many solutions
- infinite many solutions
- None

37. A 3 x 3 identity matrix have three and ___ eigen values

38. Numerical methods for finding the solution of the system of equations are classified as direct and ___ methods

- Indirect
- Iterative ✔
- Jacobi
- None

39. Which of the following is a forward difference table for the given values of x and y?

x 0.1 0.7 1.3

y 0.003 0.248 0.697

x | y | Δy | Δ^{2}y |

0.1 | 0.003 | 0.245 | 0.204 |

0.7 | 0.248 | 0.449 | | ✔ |

1.3 | 0.697 | | |

40. While using Relaxation method, which of the following is the Residuals for 1st iteration on the system; 2x + 3y = 1, 3x + 2y = 4?

- (2, 3)
- (3, -2)
- (-2, 3)
- (1, 4) ✔