Let be the set of all matrices with entries from Let be the identity matrix and the characteristic polynomial of a square matrix .

**Problem**. Let and suppose that and Prove that

**Solution**. We’ll consider three cases:

*Case 1*. *and * *is invertible *: in this case we’ll have Thus and are similar and hence

*Case 2*. *and* *is not invertible *: in this case, since the equation has a finitely many solutions for there exists some such that for all i.e. is invertible for all Thus, by *Case 1*, for any with , we have

.

Now letting we’ll get

*Case 3*. : in this case we add zero rows (resp. columns) to (resp. ) to get an matrix (resp. ). It follows immediately that

and .

Therefore:

Thus, by *Case 1 *and* Case 2*

**Remark**. So if , then, counting multiplicity, the eigenvlaues of and are the same. If counting multiplicity, the eigenvalues of are the same as the eigenvalues of plus eigenvalues which are all .

Case2, you already assume that 0<t0′ ?

As I mentioned in my post,