Gauss-Jordan Elimination
What is Gauss-Jordan Elimination?
What is Gauss-Jordan Elimination?
Gauss-Jordan Elimination or Gaussian Elimination is an algorithm to solve a system of linear equations and find the inverse of a invertible matrix.
The main goal to create an augmented matrix and put the rows in reduced echelon form where all the numbers in the principal diagonal are 1 and everything below the principal diagonal is 0. After that we can use substitution to find the answers of our system of equations.
If we continue to reduce the augmented matrix to reduced row echelon form then the numbers in the principal diagonal are 1 and every other number is 0.
Row Operations
Guidelines For Finding the Echelon Form
1. Locate the first column with the number 1 and transform it by any of the 3 row operations most likely operation 1 or 2.
2. Apply operation 3 to get 0s under the number and the remaining rows.
3. Disregard the first row. Locate the next column that contains nonzero elements and transform it by applying any of the 3 row operations most likely 1 or 2 to get 1 in the next column and in the second row.
4. Apply operation 3 to get 0s under the number and the remaining rows.
5. Disregard the first row. Locate the next column that contains nonzero elements and transform it by applying any of the 3 row operations most likely 1 or 2 to get 1 in the next column and in the second row.
6. Continue the process until either row echelon from or reduced row echelon is formed.
Sometimes the numbers are not friendly (fractions will be involved here) and when that happens. I like to do is to leave the first nonzero element in each row as is. For instance, if it is 4, then don't multiply by 1/4 until the last. This way the fractions don't appear until the last step. The numbers might be a little bigger, but there won't be any fractions.