An infeasible linear programming problem is detected in the final simplex tableau when all optimality conditions are satisfied (no negative indicators in the bottom row for a maximization problem), but at least one artificial variable remains in the basis with a strictly positive value (> 0).

An infeasible problem is detected if the final simplex tableau contains a row with a contradictory constraint (e.g., $0 =$ a nonzero value), indicating no feasible solution exists.

If an artificial variable remains positive in the final solution, the problem is infeasible.