Step 1: Represent the rotation using a complex exponential.
A rotation matrix by angle $\theta$ corresponds exactly to multiplication by the complex number $e^{i\theta} = \cos\theta + i\sin\theta$ on the complex plane, if we think of a vector $(x,y)$ as the complex number $x + iy$. Multiplying by $M$ once is the same as multiplying by $e^{i\theta}$ once.
Step 2: Translate $M^{2026}$ into this complex number language.
Applying $M$ a total of $2026$ times corresponds to multiplying by $e^{i\theta}$ a total of $2026$ times, which is:
\[ \left(e^{i\theta}\right)^{2026} = e^{i \cdot 2026\theta} \]
Step 3: Reduce the exponent modulo $2\pi$.
Since $\theta = \dfrac{2\pi}{5}$, we get $2026\theta = 2026 \times \dfrac{2\pi}{5}$. A complex exponential $e^{i\phi}$ repeats with period $2\pi$ in $\phi$, that is, $e^{i(\phi + 2\pi k)} = e^{i\phi}$ for any integer $k$.
So we only need $2026\theta \bmod 2\pi$. Because $\theta = \dfrac{2\pi}{5}$, this is equivalent to reducing $2026 \bmod 5$.
$2026 = 5(405) + 1$, so $2026 \equiv 1 \pmod 5$.
Step 4: Simplify.
\[ e^{i \cdot 2026\theta} = e^{i(405 \cdot 2\pi + \theta)} = e^{i \cdot 405 \cdot 2\pi} \cdot e^{i\theta} = 1 \cdot e^{i\theta} = e^{i\theta} \]
Since $e^{i \cdot 405 \cdot 2\pi} = 1$ ($405$ full turns bring us back to the start).
Step 5: Translate back to matrix form.
$e^{i \cdot 2026\theta} = e^{i\theta}$ means $M^{2026}$ is the same rotation as $M^1$, so:
\[ M^{2026} = M \]
\[ \boxed{M^{2026} = M} \]