Show that $\lim\limits_{t \to 0}\frac{Det(A+tB)-Det(A)}{t} = Det(A)\cdot Trace(A^{-1}B)$. Also show that $\langle \nabla Det(A),B \rangle=Det(A)\cdot Trace(A^{-1}B)$.
I know that $det(tI_{n\times n}+C)=t^{n}+Trace(C)t^{n-1}+...+Det(C)$.
This is what I've tried so far: $$\lim\limits_{t \to 0}\frac{Det(A+tB)-Det(A)}{t} \cdot Det(A^{-1})\cdot Det(A)$$ $$\lim\limits_{t \to 0}\frac{Det(I_{n \times n}+tBA^{-1})-1}{t} \cdot Det(A)$$ Let $C=tBA^{-1}$: $$\lim\limits_{t \to 0}\frac{Trace(C)\cdot 1^{n-1}+...+Det(C)}{t} \cdot Det(A)$$ But this doesn't seem to be the correct path to the solution.