The standard approach to error analysis of linear systems is to consider that the given system represents any of the systems
(A + ΔA) * (x + Δx) = b + Δb
where ΔA and Δb have entries of relative size μ = 5 * 10-d, so that
||ΔA|| ∼ μ * ||A|| and ||Δb|| ∼ μ * ||b||.
The idea being that the solution found will represent the exact solution of a perturbed system with perturbations in the bounds given.
By standard manipulations of truncated geometric or Neumann series
(A + ΔA) * Δx = Δb - ΔA * x
and ignoring all second order terms,
Δx ≃ A-1 * Δb - A-1 * ΔA * x = A-1 * Δb - A-1 * ΔA * A-1 * b
so that
||Δx|| ≃ ||A-1|| * ||Δb|| + ||A-1|| * ||ΔA|| * ||x|| ≦ μ * (||A-1|| * ||b|| + κ * ||x||)
||Δx|| ≦ 2 * μ * κ * ||x||
The relative error of x, ||Δx||/||x||, which will determine the number of valid digits in x, is about or smaller than 2 * μ * κ. Per assignment, this has to be smaller than 5 * 10-e, or
2 * κ ≦ 10d-e.
and with the specific formula for κ,
2 * λ * nα ≦ 10d-e.