Consider a 3x3 matrix A whose eigenvalues are +1, +2, and -3 and whose eigenvectors are e1, e2, and e3.
Suppose that we construct a new matrix using the formula:
$B = A^3 + I$
Q: What are the eigenvalues of B? Do not attempt to reconstruct the matrix A or guess at eigenvectors. Work out your answer
using what we know about the properties of eigenvectors and eigenvalues and matrix powers.