Determinant as a measure of instability of a linear system

Question

Suppose we have a system of linear algebraic equations defined by $Ax=B$, where $A$ is $n\times n$ real matrix and $B$ is $n\times 1$ column. Judging by the $\det A$ how can we determine the extent of how stable the given system is? For example, if $\det A<10^{-p}$ can we conclude that the system is susceptible to changes in $k(p)$-th digit after the decimal point, or that changes after $k_1(p)$-th digit after the decimal would entail changes in solution after $k_2(p)$-th digit? What would be the rule for $k_{1,2}(p)$ here?

Answer 1

It's dangerous to use a determinant as a measure of condition number for your system.
Scaling the matrix A does not change the condition number of the system while it does change the determinant of A! In fact, determinants are only a measure for the absolute pertubartion of the solution. Therefor, no general conclusion about how many decimals will be lost, can be made.

NB: be careful not to confuse the conditioning of a problem with the stability of an algorithm.