Several Linear Equations with MATLAB

Suppose we want to solve the problem

where A_ij is a square nxn matrix, f_i is a vector of length n. x_i is a vector of length n, whose elements we wish to determine. Take the example with the matrix A_ij and vector f_i as shown.

Next consider how the problem is solved in MATLAB. The matrix is entered by using the following command.

>>A = [3.2 4.1
-5.0 0.8]
A =
3.2000 4.1000
-5.0000 0.8000

The vector f is entered using the following command.

>>f = [3
4]
f =
3
4

The solution to the problem is found using:

>>x=A\f
x =
-0.60711188204683
1.20555073720729

(Use format long, and then >> x to get the long version of x.) To test the accuracy we calculate the vector that should be zero.

>>y = A*x - f
y = 1.0e-15 *

-0.44408920985006
0

The elements of y are less than 10^-15 in one case and zero to machine accuracy in the other. The solution is very accurate.

If we had entered f as a row vector instead of a column vector, i.e. as

f = [3 4]

then upon using

x = A\f

we would have received this message.

??? Error using ==> \
Matrix dimensions must agree.

We then take the transpose of f, and use the command

x = A\f'

to obtain the correct result.