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  2. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    Now through application of elementary row operations, find the reduced echelon form of this n × 2n matrix. The matrix A is invertible if and only if the left block can be reduced to the identity matrix I; in this case the right block of the final matrix is A −1. If the algorithm is unable to reduce the left block to I, then A is not invertible.

  3. LU decomposition - Wikipedia

    en.wikipedia.org/wiki/LU_decomposition

    LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

  4. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Matrix P represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I , so an LU decomposition exists.

  5. Crout matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Crout_matrix_decomposition

    The Crout matrix decomposition algorithm differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout method returns a lower triangular matrix and a unit upper triangular matrix. So, if a matrix decomposition of a matrix A is such that: A = LDU

  6. Row echelon form - Wikipedia

    en.wikipedia.org/wiki/Row_echelon_form

    A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it.

  7. 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.

  8. Today's Wordle Hint, Answer for #1271 on Wednesday, December ...

    www.aol.com/todays-wordle-hint-answer-1271...

    If you’re stuck on today’s Wordle answer, we’re here to help—but beware of spoilers for Wordle 1271 ahead. Let's start with a few hints.

  9. Pivot element - Wikipedia

    en.wikipedia.org/wiki/Pivot_element

    A pivot position in a matrix, A, is a position in the matrix that corresponds to a row–leading 1 in the reduced row echelon form of A. Since the reduced row echelon form of A is unique, the pivot positions are uniquely determined and do not depend on whether or not row interchanges are performed in the reduction process.