When calculating $2P$ where $P = (7, 11)$ on the elliptic curve E: $y^2 = x^3 + x + 1 \mod 23$.
I get $$ \lambda = \frac{3 * 49 + 1}{2 * 22} = \frac{74}{11} \mod \ 23 = 10.$$
Then when I calculate $ x_3 = \lambda^2 - 2 * x_1 = (10^2 - 14) \mod \ 23 = 17$.
and $ y_3 = -y_1 + \lambda \ (x_1 - x_3) = (-11 + 10 \ (7 - 17)) \mod \ 23 = 4$
The calculated coordinates for 2P turned out to be (17, 4), which does not seem to lie on the elliptic curve. I would appreciate any insights into what might be the problem with this calculation.
You made an arithmetic mistake for the slope $$. Since $11^{−1}≡−2 \bmod{23}$, then $74/11≡5⋅(−2)=−10≡13 \bmod{23}$.