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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