Is $det(A)=0$ a good indicator to say that a matrix is not invertible? In finite elements, for example, appears huge sparce (CRS) matrices (matrices with a lot of zeros). It is possible that matlab (or some other program) calculates $det(A)=0$ even though the matrix is invertible?

Absolutely.
There’s always round off error, and the numerical stability of algorithms for calculating determinates can be highly unstable. 
There are numberical techniques to find inverses for sparse matrices - I don’t know any, but Google will.
