1. **State the problem:** Find the eigenvalues of the matrix $$\begin{bmatrix}4 & 1 \\ 2 & 3\end{bmatrix}$$. 2. **Formula:** 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:** $$A - \lambda I = \begin{bmatrix}4 - \lambda & 1 \\ 2 & 3 - \lambda\end{bmatrix}$$ 4. **Calculate the determinant:** $$\det(A - \lambda I) = (4 - \lambda)(3 - \lambda) - (2)(1)$$ 5. **Expand and simplify:** $$= (4)(3) - 4\lambda - 3\lambda + \lambda^2 - 2 = 12 - 7\lambda + \lambda^2 - 2 = \lambda^2 - 7\lambda + 10$$ 6. **Solve the quadratic equation:** $$\lambda^2 - 7\lambda + 10 = 0$$ 7. **Factor the quadratic:** $$ (\lambda - 5)(\lambda - 2) = 0 $$ 8. **Find eigenvalues:** $$\lambda = 5 \quad \text{or} \quad \lambda = 2$$ **Final answer:** The eigenvalues of the matrix are $5$ and $2$.