1. **State the problem:** Find the eigenvalues of the matrix $$\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$$. 2. **Formula used:** Eigenvalues $\lambda$ satisfy the characteristic equation $$\det(A - \lambda I) = 0,$$ where $A$ is the matrix and $I$ is the identity matrix. 3. **Set up the characteristic matrix:** $$A - \lambda I = \begin{bmatrix} 2 - \lambda & 1 \\ 1 & 2 - \lambda \end{bmatrix}$$ 4. **Calculate the determinant:** $$\det(A - \lambda I) = (2 - \lambda)(2 - \lambda) - (1)(1) = (2 - \lambda)^2 - 1$$ 5. **Expand and simplify:** $$ (2 - \lambda)^2 - 1 = (4 - 4\lambda + \lambda^2) - 1 = \lambda^2 - 4\lambda + 3$$ 6. **Solve the quadratic equation:** $$\lambda^2 - 4\lambda + 3 = 0$$ Using the quadratic formula: $$\lambda = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2}$$ 7. **Find the eigenvalues:** $$\lambda_1 = \frac{4 + 2}{2} = 3$$ $$\lambda_2 = \frac{4 - 2}{2} = 1$$ **Final answer:** The eigenvalues of the matrix are $$\boxed{3 \text{ and } 1}$$.