Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same set of unknowns, typically expressed in matrix form as [A]{x} = {b}. The nature of the solution depends on the properties of the coefficient matrix [A]. A unique solution exists if and only if [A] is square and its determinant is non-zero (i.e., it is non-singular or invertible), which also implies the matrix has full rank. If det([A]) = 0, the system is singular and may have either no solution (inconsistent) or infinitely many solutions (dependent). Numerical methods for solving these systems are broadly classified into direct and iterative methods. Direct methods (e.g., Gaussian elimination, LU decomposition) provide an exact solution (up to round-off error) in a finite number of steps and are preferred for small to medium, dense systems. Iterative methods (e.g., Jacobi, Gauss-Seidel) start with an initial guess and generate a sequence of approximations that converge to the true solution. They are highly efficient for very large, sparse systems (common in FEA and CFD) because they preserve the zero entries and require less memory. The convergence of iterative methods is guaranteed if the coefficient matrix is strictly diagonally dominant. The condition number of a matrix, κ(A) = ||A||·||A⁻¹||, measures the sensitivity of the solution to small changes in the input data. A high condition number indicates an ill-conditioned system, where small perturbations in [A] or {b} can lead to large errors in {x}, making the numerical solution unreliable without specialized techniques like pivoting or higher precision arithmetic.