1. **Problem:** Given matrix $$A=\begin{pmatrix}3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix}$$ with eigenvalues 3 and 6, find the eigenvalues of $$A^{-1}$$. 2. **Step 1: Recall properties of eigenvalues and inverse.** The eigenvalues of the inverse matrix $$A^{-1}$$ are the reciprocals of the eigenvalues of $$A$$, provided the eigenvalues are nonzero. 3. **Step 2: Use given eigenvalues.** Eigenvalues of $$A$$ are 3, 6, and one unknown (say, $$\lambda_3$$). 4. **Step 3: Find the third eigenvalue using the trace.** The trace (sum of diagonal elements) of $$A$$ is: $$\operatorname{tr}(A) = 3 + 5 + 3 = 11$$ The sum of eigenvalues equals the trace: $$3 + 6 + \lambda_3 = 11$$ Solving: $$\lambda_3 = 11 - 9 = 2$$ 5. **Step 4: Eigenvalues of $$A$$ are 3, 6, and 2.** 6. **Step 5: Compute eigenvalues of $$A^{-1}$$.** The eigenvalues of $$A^{-1}$$ are: $$\frac{1}{3}, \frac{1}{6}, \frac{1}{2}$$ 7. **Final answer:** Eigenvalues of $$A^{-1}$$ are $$\boxed{\left\{\frac{1}{3}, \frac{1}{6}, \frac{1}{2}\right\}}$$.