Gaussian elimination turns matrices into staircases of zeros
Row operations—swap two rows, scale a row, or add a multiple of one row to another—systematically clear entries beneath pivots. Named for Carl Friedrich Gauss, the algorithm solves linear systems and also yields rank, determinants, and matrix inverses.
Start with the coefficient matrix of a linear system, or an augmented matrix that carries the right-hand side along. Elementary row operations never change the solution set, yet they can force the lower-left corner toward zeros. Forward elimination reaches row echelon form: nonzero rows above zero rows, each leading entry to the right of the one above, zeros below pivots. From that triangular shape you can already see whether there are no solutions, one solution, or infinitely many. Back substitution—or continued reduction to reduced row echelon form—finishes the solve. Stopping at echelon form is often cheaper; pushing to the unique reduced form is Gauss–Jordan elimination.
Reduced row echelon form demands leading ones and zeros elsewhere in those pivot columns. That final shape does not depend on which legal sequence of row operations you chose. Algebraically, each elementary operation is left-multiplication by an elementary matrix; packages of them give Frobenius matrices. Forward elimination corresponds to an LU factorization viewpoint, while the full reduction writes the original matrix as an invertible factor times a uniquely determined reduced matrix.
Pivots are the leftmost nonzero entries of rows; when two compete in a column, a type-3 operation clears one, and swaps restore the echelon order. The same toolkit computes the rank (number of nonzero rows in echelon form), the determinant of a square matrix (via the product of pivots, up to sign from swaps), and the inverse of an invertible matrix (by reducing alongside an identity block). From hand calculation to numerical linear algebra libraries, clearing below the diagonal remains the workhorse move.
Source: Gaussian elimination
