Row reduction
Elimination works on two equations by insight. Is there one routine that settles any linear system, of any size, every time?
Where you are 7 inside
What this area is about
Row reduction is elimination made mechanical. Write a linear system as an augmented matrix, apply moves that never change its solutions, and aim for a staircase shape. The last rows then say whether the system has one solution, none (it is inconsistent), or infinitely many, and give them all.
How these fit together
Definitions and moves. Augmented matrices and row operations sets up the matrix and the three moves: swap two rows, scale a row by a nonzero number, add a multiple of one row to another.
Procedures. Echelon forms and Gaussian elimination gives the staircase target and a strategy that reaches it. Solving a system by row reduction reads the three cases off the final rows.
Results. Reduced row echelon form is unique for each matrix, whatever route reaches it, so the rank, defined as its number of pivots, is a fixed number for each matrix. Solution sets write every solution of a consistent A\mathbf{x} = \mathbf{b} as one particular solution plus the solutions of A\mathbf{x} = \mathbf{0}.
Application. Linear systems in use sets up networks, chemical equations and curve fits as systems.
A path through it
Row operations, then echelon forms. From there two strands run, and they refer to each other: reduced row echelon form, then rank; and solving by row reduction, then solution sets, then applications.
Row reduction finds inverses (row reducing A next to I) and kernels (kernel), and rank feeds rank and nullity.