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  2. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].

  3. Tridiagonal matrix algorithm - Wikipedia

    en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

    In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as

  4. Schur complement - Wikipedia

    en.wikipedia.org/wiki/Schur_complement

    The Schur complement arises when performing a block Gaussian elimination on the matrix M.In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows: = [] [] [] = [], where I p denotes a p×p identity matrix.

  5. LU decomposition - Wikipedia

    en.wikipedia.org/wiki/LU_decomposition

    The above procedure can be repeatedly applied to solve the equation multiple times for different b. In this case it is faster (and more convenient) to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b, rather than using Gaussian elimination each time

  6. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Comments: The LUP and LU decompositions are useful in solving an n-by-n system of linear equations =. These decompositions summarize the process of Gaussian elimination in matrix form. Matrix P represents any row interchanges carried out in the process of Gaussian elimination.

  7. Elementary matrix - Wikipedia

    en.wikipedia.org/wiki/Elementary_matrix

    Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form .

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  9. Cramer's rule - Wikipedia

    en.wikipedia.org/wiki/Cramer's_rule

    In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. [8] [9] [verification needed] Cramer's rule can also be numerically unstable even for 2×2 systems. [10]