Gauss Elimination with partial pivoting

Hi,

I am trying to perform Gauss-Elimination with partial pivoting in MATLAB and I am unfortunately not obtaining the correct solution vector. My pivots are not getting switched correctly either. I am unsure of what the correct way of coding it in is. Please help me understand what I am doing wrong and what the correct code should look like. Thank you! Below is my code attached:

clc;

clear;

% User enters matrix

m = input('enter the number of rows '); %number of rows

% Trapping bad user inputs

while isempty(m) ||isstring(m)

m = input('\n Error! please enter again the number of rows ');

end

n = input('\n enter the number of columns '); % number of columns

while isempty(n) || isstring(n)

n = input('\n Error! please enter again the number of rows ');

end

% Initializing matrix A with m rows and n columns

A = zeros(m,n);

%Accepting user input for matrix A

for i=1:m

for j=1:n

fprintf('\n Enter the %d row and %d column ',i,j);

A(i,j) = input('\n');

end

end

% Displaying user entered matrix

fprintf('Your Matrix is \n');

disp(A);

% Accepting user input for desired vector b

b = input('Enter a vector b (must be of same dimension as matrix A) : ');

Ab = [A b];

%Process of forward elimination with partial pivoting

[max_row,idx] = max(Ab);

%partial pivoting

for l=1:n

max_row = Ab(l,:);

Ab(l,:) = Ab(idx(l),:);

Ab(idx(l),:) = max_row;

end

for k = 1:m-1

%Forward elimination

mult = A(k+1:m,k)/A(k,k); %This is a multiplier used to reduce pivot row

Ab(k+1:m,:) = Ab(k+1:m,:) - mult*Ab(k+1,:); % row reduction for matrix A

%b(k+1:m,:) = b(k+1:m,:) - mult*b(k,:); % row reduction for desired vector b

end

%initializing solution vector x

x = zeros(m,1);

% obtaining solution vector using back substitution

for k=m:-1:1

x(k) = (Ab(k,end) - Ab(k,k+1:m)*x(k+1:m))/Ab(k,k); % finding the remaining variables

end

% Displaying the reduced matrix

fprintf('The reduced matrix is \n');

disp(Ab);

% displaying solution to user

fprintf('The solution vector is \n');

disp(x);

% Using built in MATLAB functions to compute solution vector

fprintf('Now, comparing answer using built in MATLAB functions');

sol = A\b;

fprintf('The solution is \n');

disp(sol);