GaussJordansolve

Gauss-Jordan elimination is a method used to solve systems of linear equations. It is an extension of Gaussian elimination, which also solves systems of linear equations. The Gauss-Jordan method differs in that it continues the process of elimination until the matrix is in reduced row echelon form, where the leading coefficient of each row is 1 and all other entries in the column are 0. This results in a matrix that directly provides the solution to the system of equations.

The process begins by forming an augmented matrix that includes the coefficients of the variables and the constants from the equations. The method then proceeds by performing row operations to transform the matrix into reduced row echelon form. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another.

Once the matrix is in reduced row echelon form, the solutions to the system of equations can be read directly from the matrix. Each row corresponds to an equation, and the variables are represented by the columns. If the system has a unique solution, the matrix will have a single row for each variable. If the system has infinitely many solutions, the matrix will have a row of all zeros, indicating a free variable. If the system has no solution, the matrix will have a row that contradicts the others, typically indicated by a row of zeros with a non-zero constant term.

The Gauss-Jordan method is particularly useful for solving systems of linear equations with multiple variables, as it provides a systematic approach to finding the solutions. It is also useful for understanding the concepts of linear independence and the rank of a matrix.