# 统计代写|主成分分析代写Principal Component Analysis代考|”Cramer’s Rule and Gauss Elimination for Solving Linear Equations”

The provided text thoroughly explains Cramer’s rule and its application to systems of linear equations, as well as the Gauss elimination method, which is a more efficient alternative for solving larger systems.

Cramer’s Rule: Cramer’s rule is a technique for solving systems of linear equations with the same number of equations as unknowns. It expresses the solution vector {x} as ratios of determinants. For a 2×2 system as shown in (C.1) and (C.2), the solutions x1 and x2 are given by (C.4) and (C.5), respectively, where the denominators are the determinant of the coefficient matrix [A]. Each numerator is the determinant of a new matrix [A1] or [A2], formed by replacing the corresponding column in [A] with the constant vector {f}.

For a general nxn system (C.9), Cramer’s rule states that the solution component xi is the determinant of the matrix formed by replacing the ith column of [A] with {f}, divided by the determinant of [A].

Determinant Condition: If the determinant of [A] is zero (|A|=0), Cramer’s rule does not provide unique solutions unless the system is homogeneous ({f}=0). In the homogeneous case, a non-zero determinant implies no non-trivial solutions, while a zero determinant suggests an infinite number of solutions lying on a lower-dimensional manifold determined by the relationship among the coefficients, as seen in (C.13).

Gauss Elimination Method: The Gauss elimination method (also called Gaussian elimination) converts the system (C.14a) into an upper triangular form (C.14b) through a series of row operations. The pivot element (a diagonal entry) is used to eliminate variables from the rows below it. The process is iterative, repeating for each column until the matrix is in upper triangular form. Solutions are then found by back substitution, starting from the last equation where the solution is simply the ratio of the last entry in the right-hand side vector to the last diagonal entry.

The algorithm outlined in the text demonstrates how to eliminate x1 from the subsequent equations by multiplying the first row by appropriate scalars and subtracting them from the respective rows. This process continues for each variable until the matrix is upper triangular.

Efficiency and Implementation: While Cramer’s rule provides a theoretical framework for solving linear systems, it can be computationally expensive due to the necessity of calculating multiple determinants, particularly for large matrices. In contrast, the Gauss elimination method is more efficient and suitable for computer implementation, requiring fewer arithmetic operations and avoiding direct computation of matrix inverses. Its efficiency can be further improved when dealing with symmetric matrices common in finite element analysis and other numerical methods.

### MATLAB代写

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