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L is a field for which [tex] a \in L [/tex]. The matrix

[tex]

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)

[/tex]

has the characteristic polynomial

[tex]

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

[/tex]

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