all.
Let A be a real symmetric $(N\times N)$ matrix.
Although I would like to check its rank and determinant in order to calculate the inverse of A, a confliction arised. $\\$ Since A is a large matrix, (I wish I could break the matrix in several small pieces and have a look), I checked the rank and the determinant through MATLAB.
Though the rank is N, the determinant is equal to 0.
>> rank(A)
ans = N
>> det(A)
ans = 0
>> cond(A)
ans = 5.2e+05
As far as I know that the full rank is identical to be invertable, but it cannot since the determinant is zero. How it can be resolved? Thank you in advance.
The condition number of a matrix, cond(A), measures the sensitivity to changes in the input. Since your matrix has a large condition number, the true value of the determinant could be very different than 0. (Or possibly the matrix may not be full rank). See the Matlab Documentation for more details.