Linear Systems: Master Numerical Methods with Gauss Elimination, LU Decomposition, and Iterative Methods

Question 1

Which of the following is NOT a step in the Gauss elimination method?

Accepted Answer

Matrix inversion

Answer

Forward elimination

Answer

Backward substitution

Answer

Row operations

Question 2

In LU decomposition, what does the U matrix represent?

Accepted Answer

Upper triangular matrix

Answer

Lower triangular matrix

Answer

Diagonal matrix

Answer

Identity matrix

Question 3

What is the stopping criterion for iterative methods?

Accepted Answer

When the error between successive solutions is below a threshold

Answer

When the solution does not change significantly

Answer

When the number of iterations reaches a maximum

Answer

When the matrix is diagonally dominant

Question 4

Identify the diagonally dominant matrix.

Accepted Answer

[(2, 1, 0), (1, 2, 1), (0, 1, 2)]

Answer

[(1, 2, 3), (4, 5, 6), (7, 8, 9)]

Answer

[(2, 0, 1), (0, 2, 0), (1, 0, 2)]

Answer

[(1, 1, 1), (1, 1, 1), (1, 1, 1)]

Question 5

What is the computational complexity of Gauss elimination for an (n x n) matrix?

Accepted Answer

O(n^3)

Answer

O(n)

Answer

O(n^2)

Answer

O(2^n)

Question 6

What is the purpose of row operations in Gauss elimination?

Accepted Answer

Transform the matrix into an upper triangular matrix

Answer

Find the inverse of the matrix

Answer

Calculate the determinant of the matrix

Answer

Solve for the eigenvalues of the matrix

Question 7

What is the first step in performing Gauss elimination?

Accepted Answer

Subtract a multiple of one row from another to create a zero below the pivot

Answer

Multiply a row by a non-zero constant

Answer

Swap the order of two rows

Answer

Create a new row by adding two existing rows

Question 8

Which representation is a valid LU decomposition?

Accepted Answer

A = U * L

Answer

A = LU

Answer

A = U + L

Answer

A = L* U

Question 9

What is the primary objective of using an iterative method to solve linear systems?

Accepted Answer

Approximating a solution with increasing accuracy

Answer

Obtaining an exact solution in a finite number of steps

Answer

Establishing the existence of a solution

Answer

Verifying the validity of a known solution

Question 10

Which methods can be employed to calculate the inverse of a matrix?

Accepted Answer

Both (A) and (C)

Answer

Gauss-Jordan elimination

Answer

Adjugate method

Answer

LU decomposition

Question 11

How do consistent and inconsistent systems of linear equations differ?

Accepted Answer

A consistent system has a unique solution while an inconsistent system has no solution

Answer

A consistent system has infinitely many solutions while an inconsistent system has no solution

Answer

A consistent system has no solution while an inconsistent system has infinitely many solutions

Answer

There is no difference between a consistent and an inconsistent system

Question 12

Which preconditioning technique is commonly used to improve the convergence speed of the Gauss-Seidel iterative method?

Accepted Answer

SOR (Successive Over-Relaxation)

Answer

Jacobi

Answer

LU Decomposition

Question 13

Which of the following is NOT a method used specifically to solve systems of linear equations?

Accepted Answer

Method of undetermined coefficients

Answer

Gauss elimination

Answer

LU decomposition

Answer

Cramer's rule

Question 14

Which of the following is NOT a characteristic of linear systems?

Accepted Answer

Linear systems can only be solved using analytical methods.

Answer

Linear systems can be solved using matrix operations.

Answer

If the coefficient matrix is invertible, the solution to a linear system is unique.

Answer

Linear systems can be used to model real-world problems.

Question 15

In a system of linear equations with two variables, what is its geometric representation?

Accepted Answer

A set of points forming lines.

Answer

A closed shape with straight sides.

Answer

A single point on a graph.

Answer

A curved shape like a parabola.

Question 16

Which condition ensures that a system of linear equations has a unique solution?

Accepted Answer

The coefficient matrix is invertible.

Answer

The number of variables is greater than the number of equations.

Answer

The system is consistent.

Question 17

Which statement about the Gauss-Seidel method is accurate?

Accepted Answer

It is an iterative method that generally converges faster than the Jacobi method.

Answer

It is a direct method that always produces the exact solution.

Answer

It is only applicable to diagonally dominant matrices.

Question 18

Which of the following methods is used to solve a system of linear equations by converting it into an upper triangular matrix, making it easier to solve by forward substitution?

Accepted Answer

Gauss elimination

Answer

LU decomposition

Answer

Jacobi iteration

Question 19

Which of the following is not a defining property of linear systems?

Accepted Answer

Solutions are always unique

Answer

Systems can be represented by matrices

Answer

Systems can have multiple solutions

Question 20

Which of the following methods can be employed to solve a system of nonlinear equations, where the equations are not linear in terms of the variables?

Accepted Answer

Newton-Raphson method

Answer

Gauss elimination

Answer

Jacobi iteration

Question 21

In iterative methods, the solution is considered converged when:

Accepted Answer

The residual vector's maximum absolute value is below a set tolerance.

Answer

A fixed number of iterations is reached, regardless of the error.

Answer

The solution vector's maximum absolute value is below a set tolerance.

Answer

The coefficient matrix's maximum absolute value is below a set tolerance.

Question 22

Which matrix represents a diagonal matrix?

Accepted Answer

[3 0 0; 0 -1 0; 0 0 2]

Answer

[1 2 0; 0 3 4; 0 0 5]

Answer

[1 2 3; 4 5 6; 7 8 9]

Answer

[2 3 1; 0 0 6; 7 8 0]

Question 23

What condition guarantees a unique solution for a system of linear equations?

Accepted Answer

The determinant of the coefficient matrix is non-zero.

Answer

The augmented matrix is in row echelon form.

Answer

The number of equations equals the number of unknowns.

Answer

The coefficient matrix is singular.