example - Example Project 1. Define a Matlab function...

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Unformatted text preview: Example Project 1. Define a Matlab function X=cramer(A,b) which solves the system A*X=b. You may assume that A is a square (n × n) matrix, and that b is a column vector in Rn . 2. Use your program to solve the system A*X=b with A=magic(7) and b=[1;2;3;4;5;6;7]. Solution 1. Cramer’s rule is a method for solving a nonsingular n × n system of linear equations Ax = b. The method is as follows: For j = 1, 2, . . . , n define an n × n matrix Cj by replacing the j th column of A by b. Thus if A = [a1 , a2 , . . . , an ], then C2 = [a1 , b, a3 , . . . , an ]. The value of xj in the (unique) solution to Ax = b is then the determinant of Cj divided by the determinant of A. This algorithm can be implemented as follows. function X=cramer(A,b) % % This function solves the system A*X=b using Cramer’s rule. % A must be an n x n matrix, and b must be an n x 1 vector. % [m,n]=size(A); d=det(A); X=zeros(n,1); for j=1:n, % The matrix C is obtained by replacing C=A; % the jth column of A with b. The jth C(:,j)=b; % entry in the solution vector is equal X(j,1)=det(C)/d; % to det(C)/det(A). end 2. We now apply our program to the given data. >> A=magic(7) A= 30 38 46 5 13 21 22 39 47 6 14 15 23 31 48 7 8 16 24 32 40 >> b=[1 2 3 4 5 6 7]’ 1 9 17 25 33 41 49 10 18 26 34 42 43 2 19 27 35 36 44 3 11 28 29 37 45 4 12 20 b= 1 2 3 4 5 6 7 >> cramer(A,b) ans = 0.0050 0.0050 0.0050 0.1300 0.0050 0.0050 0.0050 ...
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This note was uploaded on 12/27/2011 for the course MAS 3114 taught by Professor Olson during the Fall '08 term at University of Florida.

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example - Example Project 1. Define a Matlab function...

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