Yes, in some sense (see below) you do care about A. But back then, everyone was worried even about well-conditioned systems. As of the mid 1940s the best erro

Yes, in some sense (see below) you do care about A. But back then, everyone was worried even about well-conditioned systems. As of the mid 1940s the best error bound on cumulative roundoff error was 4^n where n is matrix order. That 4^n would be a multiplier on top of condition number (which at the time had not been formalized, it was Turing who did that). Anyway, the dependence on matrix size turned out to be order n^2, and in practice more like O(n). * * * Also, the bounds on solving A x = b for x, knowing b and A, work differently than solving for the inverse of A. Sometimes for a given A and b, you can compute x above quite accurately, but you can't compute inv(A) accurately and multiply by b to get x. It depends on the how the "difficult directions" in A interact with b. This is one reason numerical analysts always ask, "Do you really need to invert A?" Because of this, the actual condition number of A (i.e., the norm of A times that of inv(A)) does not enter into the bounds on solving A x = b. The bounds depend on a different norm. In fact, there are several forms of the error bound -- known as backward and forward errors -- depending on what you need. * * * Incidentally, this is the kind of stuff that's covered in Golub and Van Loan, or in Trefethen and Bao, but not in a standard Abstract Linear Algebra book. For instance, Strang's book spends only one chapter on numerical linear algebra.