Characteristic polynomial

1. Jan 5, 2007

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

2. Jan 6, 2007

Hurkyl

Staff Emeritus
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?

3. Jan 6, 2007

mathwonk

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

4. Jan 7, 2007

keddelove

Oops, should have stated:

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