Taylor Expansion on Determinant

[unscientific]

Homework Statement

Show by direct expansion that:

det (I + εA) = 1 + εTr(A) + O(ε²)

Homework Equations

f(x) = f(a) + (x-a)f'(a) + (1/2)(x-a)²f''(a) + ...

The Attempt at a Solution

Does the question mean Taylor expansion when they say 'direct expansion'?

I'm kind of stuck on how to differentiate a determinant.

Expanding det (I + εA) about det (I) :

det (I + εA) = det(I) + [det (I + εA) - det(I)]f'(x) + ...

 

[UltrafastPED]

I think they mean "expand the determinant" - you will find a lot of information here:
http://en.wikipedia.org/wiki/Determinant#Sylvester.27s_determinant_theorem

However most proofs are simplest with the Levi-Civita description of the determinant:
http://en.wikipedia.org/wiki/Levi-Civita_symbol#Determinants

Remember that the determinant is polynomial in the coefficients; here you have an extra parameter, ε, which will appear in almost all of the terms.