SFWRENG 3X03 Study Guide - Final Guide: Tridiagonal Matrix, Pivot Element, Row And Column Vectors
18 Oct 2016
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Problem setting: solve for x in the system of linear equations: A b x n-by-n matrix, nonsingular; n-by-1 vector; n-by-1 solution vector. Since a is nonsingular, this system has a unique solution for any right-hand-side vector b. The following section mainly discusses the methods for solving ax = b, where a is a dense matrix, so that matrix a is stored in a two dimensional array. Cramer"s rule is a standard textbook method for solving linear systems. The ith entry xi of the solution x is given by: (example question a. 1) xi = det(ai )/ det(a) where ai is the matrix formed by replacing the ith column of a by b. Cramer"s rule is only effective for small values of n such as 1, 2, or 3. Such as if a scalar a * x = b, we know that x = b/a = a-1b. Computing the inverse matrix is not efficient, but computationally simple for small n:
