Gaussian elimination

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 leading 1s have all other entries 0, and the leading 1 in each nonzero row is to the right of the leading 1 in the previous row. This final form is unique; in other words, it is independent of the sequence of row operations used. For example, in the following sequence of row operations (where two elementary operations on different rows are done at the first and third steps), the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form. [ 1 3 1 9 1 1 − 1 1 3 11 5 35 ] → [ 1 3 1 9 0 − 2 − 2 − 8 0 2 2 8 ] → [ 1 3 1 9 0 −...