Row Reduction Method

The row reduction method solves a system $AX = B$ by writing the coefficients and constants together as a single augmented matrix $[A \mid B]$, and then applying the same elementary row operations used earlier to find inverses — row interchange, scaling a row by a non-zero scalar, and adding a multiple of one row to another. The goal is to reduce the coefficient part of the augmented matrix to the identity matrix (or as close to row-echelon form as the system allows); once that's done, the constants column directly gives the value of each unknown. Because it works entirely through row operations rather than computing a determinant or an inverse fi …