I'd like to find a condition that allows me to determine if a matrix is invertible or not. naively, I computed the determinant to see if it was zero. but then I realized that even for very small values (1e-100), lapack is still able to find an invert matrix.
I read here that a $QR$ factorization can be used to compute the rank of a matrix.
Is a $QR$ factorization the best way to check if a matrix is invertible? will it take into account the numerical errors ? is there a more efficient way ?
I also found this link that gives a reciprocal condition number $\kappa$. What does this number mean ? I know that if $\kappa=\infty$ then the matrix is singular ! But what is the cutoff, what is $\infty$ for a computer ?