Taylor Expansion on Determinant

1. The problem statement, all variables and given/known data



Show by direct expansion that:



det (I + εA) = 1 + εTr(A) + O(ε²)



2. Relevant equations



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



3. 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) + ...

1. The problem statement, all variables and given/known data







2. Relevant equations







3. The attempt at a solution

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