Gaussian elimination

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In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: swapping two rows, multiplying a row by a nonzero number, and adding a multiple of one row to another row. Using these operations, a matrix can always be transformed into reduced row echelon form: each nonzero row is above every zero row, each nonzero row has leftmost nonzero entry equal to 1, the columns containing these...

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