(i)
2 * 6 = 6
6 * 1 = 1
2 * (1 * 6) = 2 * 6 = 6
6 * (2 * 3) = 6 * 3 = 6
(ii) * is commutative if for any a, b ∈ A, we have a * b = b * a. You can verify this visually by checking for symmetry in the table along the main diagonal (going from top left to bottom right).
* is not commutative because
6 * 1 = 1
while
1 * 6 = 6
(iii) If e is the identity, then for any a ∈ A, we have a * e = e * a = a.
Here, 2 is the identity element because 1 * 2 = 1 and 2 * 1 = 1, and the same is true if we replace 1 with 2, 3, or 6.
(iv) Let a ∈ A. a has an inverse a ⁻¹ if a * a ⁻¹ = a ⁻¹ * a = e.
We know that e = 2, so look for any pairs that map to 2. The table shows that
1 * 3 = 2 and 3 * 1 = 2
2 * 2 = 2
so 1 has an inverse of 3, 3 has an inverse of 1, and 2 has itself as its inverse (which is to be expected for the identity).