I think you'd best rethink your analysis. Does the following matrix satisfy the conditions?
What is the characteristic polynomial of A?
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)