The Gauss-Jordan method is an algorithm for solving systems of linear equations that involves transforming a matrix into a stepwise form. Here are the basic steps and formulas used in this method:
- Formation of the extended matrix:
- For the system of equations (Ax = b), an extended matrix ([A|b]) is formed.
- Reduction to step form:
- Use basic string operations:
- Swapping two strings.
- Multiplying a string by a non-zero number.
- Adding a multiple of the value of another row to one row.
- Normalization of leading units:
- Convert each row to a form where the leading element (the first non-zero element) is 1.
- Zeroing of supply elements:
- Use string operations to zero out all elements below the leading ones.
- Reverse substitution:
- After reducing the matrix to canonical form, the values of the variables can be easily found.
- Formulas for elementary operations:
- If (R_i) and (R_j) are the rows of the matrix, then:
- Permutation: (R_i \leftrightarrow R_j)
- Multiplication: (R_i \leftarrow k \cdot R_i) (where (k \neq 0))
- Addition: (R_i \leftarrow R_i + k \cdot R_j)
These steps will help you use the Gauss-Jordan method to solve systems of linear equations.