Chapter 7 - Eigenvalues and Eigenvectors - 7.1 The Eigenvalue/Eigenvector Problem - True-False Review - Page 443: i

True

If $\lambda$ is an eigenvalue of the matrix A, obtain $A^3v=A^2(Av)=A^2(\lambda v)=\lambda (A^2v)=\lambda (\lambda^2 v)=\lambda^3 v$ Hence, If $\lambda$ is an eigenvalue of the matrix A, then $\lambda^3$ is an eigenvalue of A^3.