Short Questions and Answers

SHORT QUESTIONS AND ANSWERS.

 

1. In what form is the coefficient matrix transformed into when AX = B is solved by Gauss-elimination method.

Solution: Upper triangular matrix.

 

2. In what form is the coefficient matrix transformed into when AX = B is solved by Gauss-Jordan method.

Solution : Diagonal matrix.

 

3. Explain briefly Gauss-Jordan iteration to solve simultaneous equations. [MU., Oct. 96]

Solution: Consider the system of equations AX = B

If A is a diagonal matrix, then the given system reduces to

This system reduces to the following n equation

Hence, we get the solution directly as

The method of obtaining the solution of the system of equations by reducing the matrix A to a diagonal matrix is known as Gauss-Jordan elimination method,

 

4. Explain the term "Pivot elements".

Solution:

The elements a₁₁, a₂₂, … a_nn which have been assumed to be non-zero are called pivot elements.

 

5. Explain the term 'pivoting'.

In the elimination process, if any one of the pivot elements a₁₁, a₂₂, ... ann vanishes or becomes very small compared to other elements in that column, then we attempt to rearrange the remaining rows so as to obtain a non-vanishing

pivot or to avoid the multiplication by a large number. This strategy is called pivoting.

Pivoting is of two types.

1. Partial pivoting,

2. Complete pivoting.

 

6. Explain the term 'partial pivoting' and 'complete pivoting'.

Problems may also arise when the pivot element is close to, rather than exactly equal to zero because if the magnitude of the pivot element is small compared to the other elements, the round-off errors can be introduced.

Therefore, before each row is normalised, it is advantageous to determine the largest available coefficient in the column below the pivot element. The rows can be switched so that the largest element is the pivot element. This is called partial pivoting. If columns as well as rows are searched for the largest element and then switched, the procedure is called complete pivoting.

 

7. Define round-off error.

The round-off error is the quantity R which must be added to the finite representation of a computed number in order to make it the true representation of that number.

 

8. The numerical methods of solving linear equations are of two types: one is direct and the other is………

Solution : Iterative

 

9. Give two direct methods to solve a system of linear equations. [A.U M/J 2012] [A.U N/D 2015 R13]

Solution: (i) Gauss elimination method, (ii) Gauss-Jordan method.

 

10. For solving a linear system, compare Gaussian elimination method and Gauss-Jordan method. [A.U. A/M 2004] [A.U. Model Paper]

Gaussian elimination method

1. Co-efficient matrix is transformed into upper triangular matrix

2. Direct method.

3. We obtain the solutions by back substitution method.

4. It is preferred for large systems of equations.

5. Less number of multiplication are required compared to Gauss-Jordon method.

Gauss-Jordan method

1. Co-efficient matrix is transformed into diagonal matrix

2. Direct method

3. No need of back substitution method.

4. It is preferred for smaller systems of equations.

5. More number of multiplications eqmo are required to Gauss dainsy elimination method.

 

11. State the principle used in Gauss-Jordan method.

Solution :

Coefficient matrix is transformed into diagonal matrix.

 

12. Solve the following system of equations by Gauss-Jordan method.

5x + 4y = 15, 3x + 7y = 12

Solution :

 

13. Solve the system of equations x-2y= 0, 2x + y = 5 by Gaussian elimination method. [A.U. M/J 2006]

Solution :

The given system is equivalent to

This is an upper triangular matrix

 [From (1) we get [by back substitution]

y = 1 and x - 2y = 0

.x – 2 = 0

x = 2.

Hence x = 2, y = 1

 

14. When does the Gauss elimination method fail?

Solution:

It fails when any one of the pivots is zero or it is a very small number, as the elimination processes. If a pivot is zero, then division by it gives over flow error, since division by zero is not defined. If a pivot is a very small number, then division by it introduces large round off errors and the solution may contain large errors.

 

15. Can we use partial pivoting in Gauss-Jordan method?

Solution: Yes.

 

16. Write the procedure involved in Gauss elimination method. [A.U M/J 2014]

Solution:

In this method, starting with the augmented matrix of the system, using elementary row operations, we transform the augmented matrix into an upper triangular matrix.

 

17. Solve, x+3y+3z = 16, x+4y+3z = 18, x+3y+4z = 19 by mala Gauss-Jordan method.

Solution:

Given: x + 3y + 3z = 16; x + 4y + 3z = 18; x + 3y + 4z = 19