What is the difference between Gauss Jordan method and the Echelon method
The Gaussian elimination method and the Gauss-Jordan method are two closely related techniques used to solve systems of linear equations and find the inverse of a matrix. Both methods involve a series of row operations, but there are some key differences between them.
The Gaussian elimination method, also known as the Echelon method, aims to transform a matrix into row-echelon form. In row-echelon form, the matrix has zeros below each leading coefficient and each leading coefficient is 1. The steps involved in the Gaussian elimination method are as follows:
1. Start with the augmented matrix of the system of equations.
2. Use row operations to create zeros below each leading coefficient (pivot) by subtracting multiples of one row from another row.
3. Continue with this process, moving from the first row to the last row, until the matrix is in row-echelon form.
4. Back-substitute to solve for the variables, starting from the last row and working your way up.
On the other hand, the Gauss-Jordan method extends the Gaussian elimination method to transform the matrix into reduced row-echelon form. In reduced row-echelon form, each leading coefficient is 1, and there are zeros both below and above each leading coefficient. The steps involved in the Gauss-Jordan method are as follows:
1. Start with the augmented matrix of the system of equations.
2. Use row operations to create zeros below and above each leading coefficient by subtracting multiples of one row from another row.
3. Continue with this process, moving from the first row to the last row, until the matrix is in reduced row-echelon form.
4. Read off the solutions directly from the matrix.
In summary, the main difference between the Gaussian elimination method (Echelon method) and the Gauss-Jordan method is that the former aims to bring the matrix into row-echelon form, while the latter brings it into reduced row-echelon form. The row-echelon form is sufficient for solving systems of linear equations, while the reduced row-echelon form provides additional information about the solution, such as the inverse of a matrix.