rækkereduktion

Rækkereduktion, a Danish term, translates to "row reduction" in English and refers to a fundamental process in linear algebra. It is a systematic method used to simplify matrices, primarily to solve systems of linear equations, determine the rank of a matrix, or find its inverse. The core technique involves applying a sequence of elementary row operations to transform a matrix into a simpler form, typically row-echelon form or reduced row-echelon form.

Elementary row operations consist of three types: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row. By applying these operations strategically, a matrix can be manipulated to isolate variables and reveal the solution set of a corresponding system of equations. For example, in solving a system of linear equations represented by an augmented matrix, row reduction aims to transform the coefficient part of the matrix into an identity matrix, making the solution readily apparent.

The process is algorithmic and can be applied to any matrix. Achieving row-echelon form means that the first non-zero element in each row (the pivot) is to the right of the pivot in the row above it, and all entries below each pivot are zero. Reduced row-echelon form further requires that all pivots are 1 and all entries above and below each pivot are zero. Rækkereduktion is a cornerstone of many computational methods in mathematics and its applications.