rowreduced

Rowreduced refers to a matrix that has undergone a sequence of elementary row operations to reach a reduced row echelon form (RREF). In many mathematical contexts, a matrix is described as row-reduced when it is in this standard form, which is a canonical representative of its row equivalence class.

A matrix in reduced row echelon form satisfies several conditions. Each nonzero row has a leading entry equal to 1, and this leading 1 is the only nonzero entry in its column. The leading 1 of a row appears to the right of the leading 1 in the row above. All zero rows are placed at the bottom of the matrix. Consequently, every pivot column contains zeros everywhere except for the pivot position.

Row reduction is accomplished by elementary row operations, typically via Gaussian elimination to reach row echelon form and then Gauss-Jordan elimination to normalize pivots and clear above them. The allowed operations are swapping two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. Importantly, the reduced row echelon form of a given matrix is unique, making it a reliable tool for classifying and solving systems.

Applications of row reduction include solving linear systems, determining the rank of a matrix, checking consistency of equations, and identifying bases for row and column spaces. The RREF directly reveals the number of free variables in a system and the structure of its solutions.

Example: For A = [ [1,2,3], [4,5,6], [7,8,9] ], the reduced row echelon form is [ [1,0,-1], [0,1,2], [0,0,0] ]. The rank is 2, and the solution set of the corresponding system can be read off from the RREF.