We denote by the set of matrices with entries from We will also denote by the adjugate of

**Remark. **If is invertible, then for all

*Proof*. Since is invertible, we have The result now follows immediately from the fact that for all

**Problem**. Suppose that are the, not necessarily distinct, eigenvalues of Prove that the eigenvalues of are

**Solution***.* We know that every element of is similar to an upper triangular matrix. So there exists an invertible matrix and an upper triangular matrix such that

So are also the eigenvalues of because similar matrices have the same characteristic polynomials. Thus the diagonal entries of are Now let Then, by (1) and the above remark

Since the diagonal entries of are using the definition of adjugate, it is easily seen that is an upper triangular matrix with diagonal entries Hence the eigenvalues of are Since, by (2), is similar to the eigenvalues of are also

**Example 1**. If the eigenvalues of are then the eigenvalues of are and

**Example 2**. If has exactly one zero eigenvalue, say then has exactly one non-zero eigenvalue, which is If has more than one zero eigenvalue, then all the eigenvalues of are zero, i.e. is nilpotent.