Gaussian elimination, also known as row reduction, is an algorithm used to solve systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. The method involves performing elementary row operations on the matrix to convert it into reduced row echelon form. This process is sometimes called Gauss-Jordan elimination. The key steps of the algorithm are:

• Write the system of linear equations in matrix form.
• Use elementary row operations to convert the matrix into reduced row echelon form.
• Solve the resulting system of equations.

Gaussian elimination can also be used to estimate the rank of a given matrix, the determinant of a square matrix, or the inverse of an invertible matrix. The method was first described in the Chinese mathematical text "Chapter Eight: Rectangular Arrays" of "The Nine Chapters on the Mathematical Art" and was later commented on by Liu Hui in the 3rd century. The name "Gaussian elimination" is used because it is a variation of Gaussian elimination as described by Wilhelm Jordan in 1888.