Gaussian elimination is a method for solving matrix equations of the form A x = b. To perform Gaussian elimination starting with the system of equations [a_11 | a_12 | ... | a_(1k) a_21 | a_22 | ... | a_(2k) ⋮ | ⋮ | ⋱ | ⋮ a_(k1) | a_(k2) | ... | a_(k k)][x_1 x_2 ⋮ x_k] = [b_1 b_2 ⋮ b_k], compose the "augmented matrix equation" [a_11 | a_12 | ... | a_(1k) a_21 | a_22 | ... | a_(2k) ⋮ | ⋮ | ⋱ | ⋮ a_(k1) | a_(k2) | ... | a_(k k) auto right match left bracketing bar b_1 b_2 ⋮ b_k][x_1 x_2 ⋮ x_k].

augmented matrix | condensation | echelon form | elementary row and column operations | Gauss-Jordan elimination | LU decomposition | matrix equation | square root method