How does casting out elevens complement the nines check?

It checks the calculation mod 11 using the alternating digit sum, and it does catch most digit transpositions that nines misses.

Because 10 ≡ −1 (mod 11), a number's remainder mod 11 is its alternating digit sum:

• 1234 → 4−3+2−1 = 2; 1324 → 4−2+3−1 = 4. Different! So elevens spots the swap nines couldn't.

You run it the same way (reduce each operand mod 11, apply the operation, compare). Its own blind spot is errors that are exact multiples of 11 — so using both the nines and elevens checks together catches almost everything.