If is a n-rowed square matrices with real entries and satisfies the equation ,then compute value of .

I tried it out like this in two ways:

1.If given equation in is its characterstic equation then (Product of eigenvalues) which gives the value of ,which is the answer.

2.But is there some elementary explanation which does not use the concept of characterstic equation.I tried to do like this:

Thus, must be even since is a square matrix with real entries

or

or

How do I rule out the possibility of