Hi.

I was asked to find the characteristic polynomial of A.

A has the following attributes:

I argue that there is a contradiction here!

if , it means that: .

if then it means that the diagonal matrix D (who has the eignvalues on it's main diagonal), has some in its entry, and the rest are zeroes. meaning, D looks like this:

now, the problem is, that , and since the trace is the sum of the eignvalues, we'll get that , and that means that ! (and not 1 as stated in the question)