A= [1 2 3;3 4 5;5 6 7],
B=[1;1;1].
I need to solve the equation $AX=B$. Here am using Matlab code like
X=linsolve(A,B).
But, using this a warning is occurred...
"Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND = 1.541976e-18."
How to correct it?
(1) First and easiest way to solve it: NOT using Matlab for this easy linear system that you can solve with a pencil and a piece of paper. :-) For instance, by means of Gaussian elimination
$$ \begin{pmatrix} 1 & 2 & 3 & | & 1 \\ 3 & 4 & 5 & | & 1 \\ 5 & 6 & 7 & | & 1 \end{pmatrix} \longrightarrow \dots \longrightarrow \begin{pmatrix} 1 & 0 & -1 & | & -1 \\ 0 & 1 & 2 & | & 1 \end{pmatrix} $$
Which tells you that the solution set of the system are all vectors of the form $(-1, 1, 0) + z (1, -2, 1)$.
(2) Secondly, if you're lazy to do those previous calculations, you can even ask Matlab to do them for you: the command "rref" reduces matrices to row-echelon form:
A = [1 2 3; 3 4 5; 5 6 7]
A =
1 2 3
3 4 5
5 6 7
b = [1 1 1]'
b =
1
1
1
Ab = [A b]
Ab =
1 2 3 1
3 4 5 1
5 6 7 1
rref(Ab)
ans =
1 0 -1 -1
0 1 2 1
0 0 0 0
(3) The reason why Matlab doesn't like this system is because its matrix $A$ is not full rank (you can see that row of zeros appearing over there). Hence, the system could have no solution at all (if we hadn't that $0$ on the fourth column, third row); or, as it happens in the present situation, have an infinite number of them. Despite of this, Matlab tries hard and finds a solution, but warning you that, since the matrix $A$ is singular, "results may be inaccurate".
X = A\b Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.850372e-18.
X =
-4.0000 7.0000 -3.0000
(4) So, it's always good to know in advance which kind of system have you got: the Rouché-Capelli theorem (also known as "Rouché-Frobenius") tells you everything you need to know about linear systems. And Matlab can help you to do the computations:
rank(A)
ans =
2
rank(Ab)
ans =
2
This tells you that the solution set is not empty and it's a straight line (dimension of the solution set $= 3 - \mathrm{rank}\ A = 1$).
