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  2. Augmented matrix - Wikipedia

    en.wikipedia.org/wiki/Augmented_matrix

    Consider the system of equations + + = + + = + + = The coefficient matrix is = [], and the augmented matrix is (|) = []. Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are an infinite number of solutions.

  3. Rouché–Capelli theorem - Wikipedia

    en.wikipedia.org/wiki/Rouché–Capelli_theorem

    The rank of a matrix is the number of nonzero rows in its reduced row echelon form. If the ranks of the coefficient matrix and the augmented matrix are different, then the last non zero row has the form [ 0 … 0 ∣ 1 ] , {\displaystyle [0\ldots 0\mid 1],} corresponding to the equation 0 = 1 .

  4. 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].

  5. Rank (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Rank_(linear_algebra)

    One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations. According to the Rouché–Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If on the other hand, the ranks of these two ...

  6. Row and column spaces - Wikipedia

    en.wikipedia.org/wiki/Row_and_column_spaces

    The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. [9] The rank of a matrix is also equal to the dimension of the column space.

  7. Overdetermined system - Wikipedia

    en.wikipedia.org/wiki/Overdetermined_system

    The rank of this matrix is 2, which corresponds to the number of dependent variables in the system. [2] A linear system is consistent if and only if the coefficient matrix has the same rank as its augmented matrix (the coefficient matrix with an extra column added, that column being the column vector of constants). The augmented matrix has rank ...

  8. Sherman–Morrison formula - Wikipedia

    en.wikipedia.org/wiki/Sherman–Morrison_formula

    A matrix (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix (in this case +) if and only if = =. We first verify that the right hand side ( Y {\displaystyle Y} ) satisfies X Y = I {\displaystyle XY=I} .

  9. Interior-point method - Wikipedia

    en.wikipedia.org/wiki/Interior-point_method

    An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. [1] The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, [2] which runs in provably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...