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  2. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    A system of equations whose left-hand sides are linearly independent is always consistent. Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ...

  3. Overdetermined system - Wikipedia

    en.wikipedia.org/wiki/Overdetermined_system

    Any system of linear equations can be written as a matrix equation. The previous system of equations (in Diagram #1) can be written as follows: [] [] = [] Notice that the rows of the coefficient matrix (corresponding to equations) outnumber the columns (corresponding to unknowns), meaning that the system is overdetermined.

  4. Coefficient matrix - Wikipedia

    en.wikipedia.org/wiki/Coefficient_matrix

    By the Rouché–Capelli theorem, the system of equations is inconsistent, meaning it has no solutions, if the rank of the augmented matrix (the coefficient matrix augmented with an additional column consisting of the vector b) is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal ...

  5. Direct stiffness method - Wikipedia

    en.wikipedia.org/wiki/Direct_stiffness_method

    The system stiffness matrix K is square since the vectors R and r have the same size. In addition, it is symmetric because is symmetric. Once the supports' constraints are accounted for in (2), the nodal displacements are found by solving the system of linear equations (2), symbolically:

  6. Rouché–Capelli theorem - Wikipedia

    en.wikipedia.org/wiki/Rouché–Capelli_theorem

    Consider the system of equations x + y + 2z = 3, x + y + z = 1, 2x + 2y + 2z = 2.. 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 infinitely many solutions.

  7. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    For example, to solve a system of n equations for n unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires n(n + 1)/2 divisions, (2n 3 + 3n 2 − 5n)/6 multiplications, and (2n 3 + 3n 2 − 5n)/6 subtractions, [10] for a total of approximately 2n 3 /3 operations.

  8. Consistent and inconsistent equations - Wikipedia

    en.wikipedia.org/wiki/Consistent_and...

    The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...

  9. 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.