# Relation of Determinant and trace of a matrix concerning derivative

by user10344621   Last Updated October 10, 2019 03:20 AM - source

If there is a function

$$F(t) = det(I_{n} + tA)$$

where

$$A$$ is an $$n \times n$$ matrix,

$$t$$ is an arbitrary real number,

and $$I_{n}$$ is $$n \times n$$ identity matrix,

is it true that the derivative of $$F(t)$$ at $$t = 0$$ is equal to the trace of $$A$$?

That is,

$$F'(0) = Tr(A)$$

I currently know that the trace is the sum of the diagonal entries of a matrix but I am not sure how I should go about differentiating the left hand side.

Any help would be appreciated.

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