1. **State the problem:** Given matrix $$A = \begin{pmatrix}34 & 26 & 20 \\ -68 & -51 & -38 \\ 32 & 23 & 16\end{pmatrix}$$, find the eigenvalues and eigenvectors of $$A^3$$ without computing $$A^3$$ explicitly. 2. **Recall the key property:** If $$\lambda$$ is an eigenvalue of $$A$$ with eigenvector $$\mathbf{x}$$, then for any positive integer $$k$$, $$\lambda^k$$ is an eigenvalue of $$A^k$$ with the same eigenvector $$\mathbf{x}$$. 3. **Find eigenvalues of $$A$$:** The eigenvalues $$\lambda$$ satisfy the characteristic equation $$\det(A - \lambda I) = 0$$. 4. **Calculate the characteristic polynomial:** $$\det\begin{pmatrix}34-\lambda & 26 & 20 \\ -68 & -51-\lambda & -38 \\ 32 & 23 & 16-\lambda\end{pmatrix} = 0$$ 5. **Simplify the determinant:** Expanding the determinant (omitted here for brevity), the characteristic polynomial is: $$f(\lambda) = -\lambda^3 -1\lambda^2 + 0\lambda + 0 = -\lambda^3 - \lambda^2$$ 6. **Solve characteristic equation:** $$-\lambda^3 - \lambda^2 = 0 \implies -\lambda^2(\lambda + 1) = 0$$ 7. **Eigenvalues:** $$\lambda_1 = 0$$ $$\lambda_2 = 0$$ $$\lambda_3 = -1$$ 8. **Find eigenvectors:** For $$\lambda = 0$$, solve $$A\mathbf{x} = 0$$: $$\begin{pmatrix}34 & 26 & 20 \\ -68 & -51 & -38 \\ 32 & 23 & 16\end{pmatrix} \mathbf{x} = \mathbf{0}$$ Row reduce to find eigenvectors: 9. **RREF for $$\lambda=0$$:** After row operations, eigenvector basis is: $$\mathbf{x}_1 = t \begin{pmatrix} -2 \\ 2 \\ -3 \end{pmatrix}$$ for scalar $$t$$. 10. **For $$\lambda = -1$$, solve $$(A + I)\mathbf{x} = 0$$: $$A + I = \begin{pmatrix}35 & 26 & 20 \\ -68 & -50 & -38 \\ 32 & 23 & 17\end{pmatrix}$$ Row reduce to find eigenvector: 11. **RREF for $$\lambda = -1$$:** Eigenvector basis: $$\mathbf{x}_3 = s \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$$ for scalar $$s$$. 12. **Eigenvalues and eigenvectors of $$A^3$$:** Eigenvalues of $$A^3$$ are $$\lambda_i^3$$: $$0^3 = 0$$ (multiplicity 2), $$(-1)^3 = -1$$. Eigenvectors remain the same. **Final answers:** - Eigenvalues of $$A^3$$: $$0, 0, -1$$ - Corresponding eigenvectors: - For $$0$$: $$\begin{pmatrix} -2 \\ 2 \\ -3 \end{pmatrix}$$ - For $$-1$$: $$\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$$