MATLAB: Residual error in Cramer's rule - matlab

Question

I am quite familiar with Cramer's rule and I know why it is "useless". However, I got curious when the residual using the code below gives something like xxxxE-016 where xxxx represent numbers. I have tried the code below with 3 to 10 matrix sizes of floating system of linear equation.
Why is the residual error minimal?
Am I missing the right syntax for disp(['norm of Cramer's rule residual = ' num2str(norm(A*x-b),4)]) in the code for getting such minimal error?
The code I provided is working perfectly, only doubting the syntax I used for the norm
function x=cramerr(A,b)
A=input('matrix A =');
b=input('vector b =');
n=size(A,1);
m=size(A,2);
tic;
if n~=m
Error ('The matrix is not square!');
x=[];
else
detA=det(A);
if det(A)~=0
x=zeros(n,1);
for j=1:n
if j~=1 & j~=n
Ab=[A(:,1:j-1) b A(:,j+1:n)];
elseif j==1
Ab=[b A(:,2:n)];
elseif j==n
Ab=[A(:,1:n-1) b];
end
x(j)=det(Ab)/detA;
end %for j=1:n
else
Error('The matrix A has a zero determinant');
x=[];
end % if det(A)~=0
toc;
disp(['norm of Cramer's rule residual = ' num2str(norm(A*x-b),4)]);
end % if n~=m
Please you can use this as an example
A =[0.373,0.296;0.326,0.260];
b =[0.521;0.456];
Thanks

Answer 1

I wouldn't say this is a problem per se. What you do when you use Cramer's rule is solve the linear system of equations. In Matlab, you can simply do that with x=inv(A)*b, which uses a different algorithm to calculate the inverse (LU decomposition). In the Matlab page on Inverses and Determinants it says:
If A is square and nonsingular, the equations AX = I and XA = I have
the same solution, X. This solution is called the inverse of A, is
denoted by A-1, and is computed by the function inv. The determinant
of a matrix is useful in theoretical considerations and some types of
symbolic computation, but its scaling and round-off error properties
make it far less satisfactory for numeric computation. Nevertheless,
the function det computes the determinant of a square matrix.
If you want to check the precision of both you could do this:
format long
built_in=inv(A)*b
own_function=r(A,b)
and compare the results. What I got was:
built_in= [1.000000000000000;0.500000000000000]
own_function= [1.000000000000054;0.500000000000000]
You can see where the small norm comes from. With this in mind such an error is most likely due to rounding off errors.
As for the formatting of disp(['norm of Cramer's rule residual = ' num2str(norm(A*x-b),4)]) I would skip the apostrophe (') in Cramer's as this will cause errors.