Singular matrix and MATLAB inversion

How come that det(A) = 0, and yet MATLAB computes the inverse of A (i.e inv(A) is computed without any warnings).

Where A is a square symmetric matrix.

Thanks !

[Adam]

You haven't shown us what A is and I don't have time to guess!

[Shivam Sharma]

let's say A = [1 2 3; 4 5 6; 7 8 9]

[Bruno Luong]

@Sharma I get the warning wth your matrix, so it's not a valide example

>> A = [1 2 3; 4 5 6; 7 8 9]:

>> inv(A)
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND =  2.202823e-18. 
 

ans =

   1.0e+16 *

    0.3153   -0.6305    0.3153
   -0.6305    1.2610   -0.6305
    0.3153   -0.6305    0.3153

Answer 1

[Steven Lord]

DON'T use det to determine if a matrix is singular!

A non-zero multiple of the identity matrix isn't singular, right?

A = 0.1*eye(400);
det(A)

The determinant of A underflowed to 0.

Can multiplying a singular (or nearly singular) matrix by a non-zero scalar make it no longer singular?

B = [1 1; 1 1+eps];
C = 1e8*B;
det(B)
det(C)

A number with a determinant on the order of 1 can't be singular, can it?

Use the cond or rcond functions instead. Matrices with condition numbers closer to 1 are better behaved.

cond(A)
cond(B)
cond(C)

If you're calling inv to try to solve a system of linear equations, DON'T. Use the backslash operator (\) instead. Even if your textbook told you to multiply by the inverse matrix, well, in theory there is no difference between theory and practice ...

[Bjorn Gustavsson]

If you have to solve a system of equations where the matrix is singular or ill-conditioned don't use backslash either. To "give" a solution in that case one should know what has been done and what has not been done. The best way to do that is to SVD-based Moore-Penrose inverse, with suitable (zeroth- second-order Tikhonov ) regularization, inspection of the singular-values, and the model-space eigenvectors etc. A great utility for this is:

regtools

Answer 2

[Bjorn Gustavsson]

Are you absolutely sure you haven't turned off the warnings, in your startup.m or elsewhere?