Gauss and GaussJordan EliminationThere are two methods of solving systems of linear equations are:
They are both based on the observation that systems of equations are equivalent if they have the same solution set and performing simple operations on the rows of a matrix, known as the Elementary Row Operations or (EROs). There are 3 EROs:
ExampleAn example of interchanging rows would be r_{1}⟵⟶r_{3}. Now, starting with this matrix, an example of scaling would be: 2r_{2}⟶r_{2},which describes all items in row 2 are multiplied by 2. Now, starting with this matrix, an example of a replacement would be: r_{3}2r_{2}⟶r_{3}.Elementbyelement, row 3, is replaced by the element in row 3 minus 2 * the corresponding items in row 2. These yields: Both the Gauss and GaussJordan methods begin with the matrix form Ax = b of a system of equations, and then augment the coefficient matrix A with the column vector b. Gauss EliminationThe Gauss Elimination method is a method for solving the matrix equation Ax=b for x. The process is:
ExampleUse a 2 x 2 system, the augmented matrix would be: Then, EROs are used to get the augmented matrix into an upper triangular form: So, it is simply to replace a_{21} with 0. Here, the primes indicate that the values have been change. Putting this back into the equation form yield Executing this matrix multiplication for each row results in: So, the solution is: Similarly, for the 3x3 system, the augmented matrix is reduced to an upper triangular form: This will be done orderly by first getting a_{0}in the a_{21} position, then a_{31}, and finally a_{32}. Then, the solution will be: Consider the following 2x2 system of equations: x_{1}+2x_{2}=2 As the matrix equation Ax = b, this is: The first process is to augment the coefficient matrix A with b to get an augmented matrix [A b]: For the forward elimination, we need to get a_{0} in the a_{21} position. To accomplish this, we can change the second line in the matrix by subtracting from it 2 * the first row. The way we would write this ERO is: Now, putting it back in the matrix equation form: says that the second equation is now 2x_{2}= 2 so x_{2} = 1. Plugging into the first equation: x_{1}+2(1)=2 This is called a backsubstitution. GaussJordan EliminationThe GaussJordan Elimination method start the similar technique that the Gauss Elimination method does, but then the instead of backsubstitution, the elimination continues. The GaussJordan method consists of:
ExampleUse 2x 2 system, the augmented matrix would be: Use 3x 3 system, the augmented matrix would be: Note: The resulting diagonal form does not include the rightmost column.For example, for the 2x2 system, forward elimination yielded the matrix: Now, to continue with back elimination, we want a_{0}in the a_{12}position. So, the solution is Here is an example of a 3x3 system: In matrix form, the augmented matrix [Ab] is Forward substitution (done orderly by first getting a_{0}in the a_{21}position, then a_{31} , and finally a_{32}): For the Gauss technique, this is followed by backsubstitution: For the GaussJordan technique, this is instead followed by back elimination: Here's an example of operating these substitutions using MATLAB:
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