Gauss elimination and Gauss-Jordan elimination are methods for solving systems of linear equations. Gauss elimination puts the matrix of coefficients into row-echelon form using elementary row operations, while Gauss-Jordan elimination further reduces the matrix to reduced row-echelon form. These methods can also be used to find the rank of a matrix, calculate the determinant, and compute the inverse of an invertible square matrix. Examples demonstrate applying the methods to solve systems of 3 equations with 3 unknowns.
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
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Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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3. System of linear equation
If all equations in a system are linear, the
system is a system of linear equation or
linear system.
Example:
2y + z = -8
x -2y -3z = 0
-x+y+2z = 3
There are two methods to solve these equations:
Direct and Iterative method
4. Gauss Elimination
Define:-
It is also known as row reduction, is an algorithm in linear algebra for solving
a system of linear equations. It is usually understood as a sequence of operations
performed on the corresponding matrix of coefficients.
This method can also be used to find the rank of a matrix, to calculate the determinant of
a matrix, and to calculate the inverse of an invertible square matrix.
History :-
The method is named after Carl Friedrich Gauss (1777–1855). Some special cases of
the method - albeit presented without proof - were known to Chinese mathematicians as
early as circa 179 CE.
5. Gauss Jorden Elimination
Gauss–Jordan elimination to refer to the procedure which ends in reduced echelon
form. The name is used because it is a variation of Gaussian elimination as described
by Wilhelm Jordan in 1888.
It is a further calculation of gauss elimination method.
For Guass elimination: [A]=
1 ∗ ∗
0 1 ∗
0 0 1
∗
∗
∗
For guass Jordan Elimination: [A]=
1 0 0
0 1 0
0 0 1
∗
∗
∗
6. Why we need these methods?
It is usually understood as a sequence of operations performed on the
corresponding matrix of coefficients. This method can also be used to
find the rank of a matrix, to calculate the determinant of a matrix, and
to calculate the inverse of an invertible square matrix.
Gaussian Elimination helps to put a matrix in row echelon form,
while Gauss-Jordan Elimination puts a matrix in reduced row echelon
form. For small systems (or by hand), it is usually more convenient to
use Gauss-Jordan elimination and explicitly solve for each variable
represented in the matrix system.
8. Guass Elimination method
First consider the system of linear equations as;
AX=B
a11x1+ a12x2+ a13x3= b1
a21x1+ a22x2+ a23x3= b2
a31x1+ a32x2+ a33x3= b3
Find the augmented matric for the given system of equations as;
C=[A;B]
C=
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
𝑏1
𝑏2
𝑏3
Use row operation to transform the augmented matrix into the Row Echelon Form.
9. An elementary row operation is;
Interchange the two rows.
Multiply a row by non-zero constant.
Add a multiple of the row to another row.
Now check the resulting matrix and re-interpret it as a system of linear
equations.
The resulting matrix will be;
1 𝑎12 𝑎13
0 1 𝑎23
0 0 1
𝑏1
𝑏2
𝑏3
Find the solution of the equations by interpreting the equations.
10. Guass Jordan method
First consider the system of linear equations as;
AX=B
a11x1+ a12x2+ a13x3= b1
a21x1+ a22x2+ a23x3= b2
a31x1+ a32x2+ a33x3= b3
Find the augmented matric for the given system of equations as;
C=[A;B]
C=
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
𝑏1
𝑏2
𝑏3
Use row operation to transform the augmented matrix into the Reduced Row
Echelon Form.
11. An elementary row operation is;
Interchange the two rows.
Multiply a row by non-zero constant.
Add a multiple of the row to another row.
Now check the resulting matrix and re-interpret it as a system of linear
equations.
The resulting matrix will be;
1 0 0
0 1 0
0 0 1
𝑏1
𝑏2
𝑏3
Find the solution of the equations.
17. Examples of Gauss Jordan
Elimination method
AMINA ZUBAIR 19011598-035
18. Example.1:
1.Solve the system of linear equations by using Gauss-Jordan Eliminating Method:
2y + z = -8
x -2y -3z = 0
-x+y+2z = 3
Solution:
The augmented matrix of the system is:
0 2 1
1 −2 −3
−1 1 2
−8
0
3
Interchanging R1 and R2
1 −2 −3
0 2 1
−1 1 2
0
−8
3
20. Example.2:
2.Solve the system of linear equations by using Gauss-Jordan Eliminating Method:
x+ y + z = 2
6x -4y+5z = 31
5x+2y+2z=13
Solution:
The augmented matrix of the system is:
1 1 1
6 −4 5
5 2 2
2
31
13
R2=R2-6R1
1 1 1
0 −10 −1
5 2 2
2
19
13
24. Applications:
Computing determinants
By using row operations of Gaussian elimination we can find out the determinant
of any square matrix
Finding the inverse of a matrix
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for
finding the inverse of a matrix.
Computing ranks and bases