There are several other methods that can be used to solve a system of linear equations besides substitution and elimination. Some of the main alternative methods include:
Gaussian elimination – This method involves performing elementary row operations on the corresponding augmented matrix of the system of equations to put it in reduced row echelon form. This is done by elimination similar to the elimination method but performing the operations on the matrix directly. The solution can then be read from the reduced row echelon form matrix. Gaussian elimination can solve systems with any number of equations and is more efficient than simple row reduction.
Matrix inverse – If the coefficient matrix of the system is invertible, we can solve the system by computing the inverse of the coefficient matrix and multiplying both sides of the equation by the inverse. For an n×n system of equations with an invertible coefficient matrix A and b as the constants vector, the solution x is given by x=A^-1b. Computing the inverse of matrices larger than 3×3 can become numerically challenging but matrix inverse directly provides the solution in one step.
Cramer’s rule – This uses the determinants of the coefficient matrix and matrices created by replacing columns of the coefficient matrix with the constants vector. For an n×n system of equations, Cramer’s rule gives the solution as x_i = det(A_i) / det(A) where A_i is the matrix with the i-th column replaced by b and A is the original coefficient matrix. Cramer’s rule provides a closed form solution but computing high dimensional determinants can become numerically unstable for large systems.
Gauss-Jordan elimination – This method extends Gaussian elimination to put the augmented matrix in reduced row echelon form with pivot entries of 1. It uses elementary row operations of interchanging rows, multiplying a row by a non-zero constant, and adding a scalar multiple of one row to another row. Gauss-Jordan puts the matrix in a simpler form than Gaussian elimination and allows easy reading of the solution.
LU decomposition – For a non-singular square system, the coefficient matrix A can be decomposed as A = LU where L is a lower triangular matrix and U is an upper triangular matrix. Then the system Ax=b can be converted to Ly = b and Ux = y and solved sequentially using forward and back substitution. LU decomposition exploits the structure of triangular matrices to efficiently solve systems.
Cholesky decomposition – If the coefficient matrix A is symmetric and positive-definite, it has a Cholesky decomposition of A = LL^T where L is a lower triangular matrix. The system can then be solved similarly to LU decomposition using only one triangular factor. Cholesky decomposition is faster than LU for positive-definite systems.
Iterative methods – For large sparse systems that are inconsistent or numerically challenging, iterative methods like Jacobi, Gauss-Seidel, SOR can be applied. These start with an initial guess and iteratively improve it using the equations until reaching the actual solution. Convergence is slower than direct methods but iterative methods can solve huge sparse systems exceeding memory limits.
That covers some of the main alternative methods for solving systems of linear equations besides the basicrow operations approach. Methods like Gaussian elimination, LU/Cholesky decomposition and matrix inverse are direct approaches providing the exact solution. Iterativemethods like Jacobi/Gauss-Seidel are applicable for special large sparse inconsistent systems. The appropriate choice depends on the size, structure and properties of the particular system being solved.