16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4.4.4

The key is recognizing that the set of all polynomials sending a matrix to zero forms the annihilating ideal of $A$, and this ideal is always principal—generated by a single monic polynomial called the minimal polynomial. You'll find it by computing the characteristic polynomial first, then checking if it annihilates $A$ or finding the smallest-degree monic polynomial that does. Since $A$ is already upper triangular, its eigenvalues are visible on the diagonal, which gives you a head start.

The key is recognizing that the set of all polynomials sending a matrix to zero forms the annihilating ideal of $A$, and this ideal is always principal—generated by a single monic polynomial called the minimal polynomial. You'll find it by computing the characteristic polynomial first, then checking if it annihilates $A$ or finding the smallest-degree monic polynomial that does. Since $A$ is already upper triangular, its eigenvalues are visible on the diagonal, which gives you a head start.