solve linear set of equations of matrix via mldivide and compute confidence intervals in matlab in matlab

To solve a linear set of equations represented by a matrix in MATLAB using the mldivide operator (\), and to compute confidence intervals, you can follow these steps:

1. Create a matrix A representing the coefficients of the linear equations.
2. Create a column vector b representing the constants on the right-hand side of the equations.
3. Use the mldivide operator (\) to solve the system of equations and obtain the solution vector x: x = A \ b.
4. Compute the confidence intervals by using the inverse of the covariance matrix of the solution vector x.

Here's an example of how to solve a linear set of equations and compute confidence intervals in MATLAB:

main.m
% Step 1: Create matrix A and vector b
A = [1, 2, 3; 2, 5, 2; 3, 7, 1];
b = [4; 3; 1];

% Step 2: Solve the system of equations
x = A \ b;

% Step 3: Compute the confidence intervals
C = inv(A' * A); % Covariance matrix
conf_intervals = tinv(0.975, size(A, 1) - size(A, 2)) * sqrt(diag(C));

% Display the solution vector x and confidence intervals
disp('Solution vector x:');
disp(x);

disp('Confidence intervals:');
disp(conf_intervals);
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In this example, A is a 3x3 matrix representing the coefficients of the linear equations, and b is a 3x1 column vector representing the constants on the right-hand side of the equations. The mldivide operator (\) is used to solve the system of equations and obtain the solution vector x. The confidence intervals are computed using the inverse of the covariance matrix C, and the tinv function is used to compute the critical value for the desired confidence level.

Note that the formula used to compute the confidence intervals assumes that the errors in the linear equations are normally distributed and independent.