GaussJordanelimination

Gauss-Jordan elimination is an algorithm for solving systems of linear equations by transforming the augmented matrix [A|b] into reduced row echelon form (RREF) using elementary row operations. If the left block of the resulting matrix becomes the identity, the right block contains the solution vector. The method also serves to determine consistency: a row of the form [0 ... 0 | c] with c ≠ 0 indicates no solution, while rows of zeros on the left indicate dependent equations and possibly infinitely many solutions. When applied to the augmented identity matrix [A|I], Gauss-Jordan elimination yields [I|A^{-1}] in the invertible case, providing the matrix inverse.

Procedure: Starting with the augmented matrix, for each column i, select a pivot by swapping rows if necessary to bring a nonzero entry into position (i,i). Scale the pivot row to make the pivot equal to 1, then use row operations to eliminate all other entries in that column. Repeat for each column. The result is a matrix in reduced row echelon form; the left block approaches the identity whenever the system is solvable with a unique solution, and the right block contains the solution or the inverse accordingly.

Relation to Gaussian elimination: Gauss-Jordan elimination extends Gaussian elimination by continuing the elimination process to also clear entries above each pivot, producing the full RREF rather than just an upper triangular form. Numerical considerations include partial pivoting to improve stability. The algorithm has a typical computational complexity of O(n^3) for an n-by-n system.

Applications: solving linear systems, computing matrix inverses, and determining matrix rank or system consistency in linear algebra and numerical analysis.