# Characteristic polynomial

#### keddelove

And again a question:

L is a field for which $$a \in L$$. The matrix

$$A = \frac{1}{2}\left( {\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ 1 & a & { - 1} & { - a} \\ 1 & { - 1} & 1 & { - 1} \\ 1 & { - a} & { - 1} & a \\ \end{array}} \right)$$

has the characteristic polynomial

$$x^4 - \left( {a + 1} \right)x^3 + \left( {a - 1} \right)x^2 + \left( {a + 1} \right)x - a$$

I need to show that this information is correct for a=1 in any field.

My problem is that when i calculate the polynomial for a=-1 I end up with another characteristic polynomial than the one given. Maybe i'm going about it the wrong way. Suggestions or pointers are very welcome

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#### Hurkyl

Staff Emeritus
Gold Member
I don't understand the question...

And if you want to show something is correct for a=1, then why are you looking at a=-1?

#### mathwonk

Homework Helper
i conjecture he meant the characteristic polynomial is accurate for a any element of any field, and yet it failed for a = -1.

#### keddelove

Oops, should have stated:

Show that this is correct for a=-1 in any field.