Question

Total(25 marks) 3. Given the system of equation as 3x + 7y - 2z=2 x - 5y + z = 13 2x + 3y - 102=-23 (a) Write a Matlab/C++ computer program to solve the system of linear equations based of the partial/scaled pivoting technique in Q3b below/ You can use any programming language] CR(10) An(9) AP(3) An(3) (b)Solve the system of equation using Gauss-Jordan Elimination method Hence Find the ii) Determinant of the matrix A, the coefficient Matrix of the system iii) The inverse A [Total: 25 marks

Answer 1

Answer 3:

• Part (a)

Matlab code:

%code to solve system of liner equations
%given equations are: 3x + 7y -2z = 2
% x - 5y + z +13
% 2x + 3y - 10z = -23
% We know AX = B, so putting the values of A and B

A = [3 7 -2; 1 -5 1; 2 3 -10];
B = [2; 13; -23];
X = linsolve(A, B) %function to get the values of x, y and z

Screenshot:

MATLAB Drive > solve.mx + 1 2 %code to solve system of liner equations *given equations are: 3x + 7y -2z = 2 x - 5y + z +13 % 2x + 3y - lez = -23 % We know AX = B, so putting the values of A and B 4 5 6 7 - 8 A = [3 7 -2; 1 -5 1; 2 3 -10]; B = [2; 13; -23]; x = linsolve(A, B) Xfunction to get the values of x, y and z 10 11 12 COMMAND WINDOW >> solve 5.0088 -1.0082 3.000 >>

• Part (b)

Matlab code:

1 5 - 6 - 9 - 12 - 13 - 14 - MATLAB Drive gauss_jordan.mx + Xcode to solve the system of equation using Gauss-Jordan method 2 function [X, er ]-gauss_jordan(A,B) A = [37 -2; 1-51; 2 3 -10] % input for augmented matrix A B = [2 ; 13; -23] % intput for matrix B [n, m]=size(A); % finding the size of matrix A er =2; % calculation of error x=zeros(n,1); % calling fuction zero if nem disp('error: nem'); % displaying error if found er = 1; 11 - end % end of the scope of if if length(B) = n % finding the legth of matrix B disp('error: wrong size of B'); % displaying error, if found er = 2; else if size(B,2) -- 1 B=B'; 18 - end % end of the scope of if-else if size(B, 2) = 1 displ'error: B is a matrix'); % displaying erron in matrix B er = 3; end end if er @ Aa=[A, B]; for i=1:n [Aa(i.n,i:n+1), er]=gauss_pivot (Aa(i:n,i:n+1));|| 28- if er == @ Aa(1:n, i.n+1)=gaussjordan_step (Aa(1:n,i:n+1),i); end end X=Aa(:, n+1); 15 - 16- 17 - 19- 20 - 21 - 22 - 23 - 26- 32 - 33 - end function Al=gaussjordan_step(A,i) % calling of fuction function 35 36

MATLAB Drive gauss_jordan.mx + - 7 end 34 - A=; 35 function Al=gaussjordan_step (A,i) % calling of fuction function 39 - 49 - 41 - 42 - 44 - 45 - 48 - 51 - 52 - 53 - [n, m]=size(A); % determination of size of matrix A A1=A; % assigning A to Al S=A1(1, 1); Al(i,:) = Ali,:)/s; k=[[1:1-1],[i+1:n]]; for j=k S=A1(0,1); A1(j, :)=A1 (j, :)-A1(i,:)*s; end % end of for loop function [A1, er]=gauss_pivot(A) % calling of fucntion [n, m]=size(A); % finding the size of matrix A Al=A; % process of assigning er = @; % error flag if A1(1,1) -- check = true; % logical(1) - TRUE i = 1; while check i = i + 1; if in disp('error: matrix is singular"); er = 1; check = false; else if Ali,1) - B && check check = false; b=A1(1,:); % process to change row i to i Al(i, :)=A1(1,:); A1(1,:)-b; end end end end 54 - 55 - 56 - 58 - 59 - ga 61 63 - 64 65 86

Source code:

%code to solve the system of equation using Gauss-Jordan method
function [x,er]=gauss_jordan_elim(A,b)
A = [3 7 -2; 1 -5 1; 2 3 -10] % input for augmented matrix A
B = [2 ; 13; -23] % intput for matrix B
[n,m]=size(A); % finding the size of matrix A
er =0; % calculation of error
x=zeros(n,1); % calling fuction zero
if n ~= m
disp('error: n~=m'); % displaying error if found
er = 1;
end % end of the scope of if
if length(B) ~= n % finding the legth of matrix B
disp('error: wrong size of B'); % displaying error, if found
er = 2;
else
if size(B,2) ~= 1
B=B';
end % end of the scope of if-else
if size(B,2) ~= 1
disp('error: B is a matrix'); % displaying erron in matrix B
er = 3;
end
end
if er == 0
Aa=[A,B];
for i=1:n
[Aa(i:n,i:n+1),er]=gauss_pivot(Aa(i:n,i:n+1));
if er == 0
Aa(1:n,i:n+1)=gauss_jordan_step(Aa(1:n,i:n+1),i);
end
end
x=Aa(:,n+1);
end
A=0;
function A1=gauss_jordan_step(A,i) % calling of fuction function

[n,m]=size(A); % determination of size of matrix A
A1=A; % assigning A to A1
s=A1(i,1);
A1(i,:) = A(i,:)/s;
k=[[1:i-1],[i+1:n]];
for j=k
s=A1(j,1);
A1(j,:)=A1(j,:)-A1(i,:)*s;
end % end of for loop
function [A1,er]=gauss_pivot(A) % calling of fucntion
[n,m]=size(A); % finding the size of matrix A
A1=A; % process of assigning
er = 0; % error flag
if A1(1,1) == 0
check = logical(1); % logical(1) - TRUE
i = 1;
while check
i = i + 1;
if i > n
disp('error: matrix is singular');
er = 1;
check = logical(0);
else
if A(i,1) ~= 0 & check
check = logical(0);
b=A1(i,:); % process to change row 1 to i
A1(i,:)=A1(1,:);
A1(1,:)=b;
end
end
end
end

To run the code, copy the source code in MatLab editor and save is as SomeFileName.m and run it. You will get the following output.

Output:

gauss_jordan.m COMMAND WINDOW 3 >> gauss_jordan A = 3 7 -2 1 -5 1 2 3 -10 B = 2 13 -23 ans = 5 -1 3

• Part (c)

(I)

1 2 solve.mx + %code to solve system of liner equations given equations are: 3x + 7y -22 = 2 x - 5y + 2 +13 2x + 3y - lez = -23 % We know AX = B, so putting the values of A and B 4 5 7 - A = [37 -2; 1 -5 1; 2 3 -10]; B = [2; 13; -23]; x linsolve(A, B) Xfunction to get the values of x, y and z %to find the determinant of matrix A Y = det(A) 10 11 12 - 13 14 15 1 COMMAND WINDOW >> solve X = 5.000e -1.2008 3.000 YE 199

(ii)

MATLAB Drive > solve.mx + A = [3 7 -2; 1-5 1; 2 3 -10]; A %to find the determinant of matrix A Y = det(A) 4 7 %to find inverse of matrix A Inverse = inv(A) 1e COMMAND WINDOW AE 3 1 2 7 -5 3 -2 1 -10 199 InverseA = 8.2362 2.0603 8.0653 0.3216 -0.1387 0.0251 -0.8151 -0.8251 -0.1106

*NOTE: Drop a comment for queries.