Book: A. Figalli 《The Monge Ampere Equation and Its Application》
1.Let $A,B\in R^{n\times n}$, and assume that $A$ is invertible. Then,
$$\frac{d}{dt}|_{t=0}det(A+tB)=det(A)tr(A^{-1}B)=tr(cof(A)^{T}B).$$
In addition, the latter formula holds also when $A$ is not invertible.
2.Let $A,B\in R^{n\times n}$, and assume that $A$ is invertible. Then,
$$\frac{d}{dt}|_{t=0}det(A+tB)^{-1}=det(A)tr(A^{-1}B)=-A^{-1}BA^{-1}.$$
3.Let $A,B\in R^{n\times n}$ be symmetric nonnegative definite matrices. Then,
$$det(A+B)\geq det(A)+det(B),$$
$$det(A+B)^{\frac{1}{n}}\geq det(A)^{\frac{1}{n}}+det(B)^{\frac{1}{n}}.$$
Furthermore, if $A,B\in R^{n\times n}$ are symmetric positive definite matrices, then
$$\log det(\lambda A+(1-\lambda)B)\geq \lambda\log det(A) +(1-\lambda)\log det(B).$$
4. Given $A\in R^{n\times n}$, we denote its operator norm by $||A||$, i.e., $||A||:=\sup_{|v|=1}|Av|$.
Assume that there exists a constant $K>1$ such that $\frac{1}{K}Id\leq A^TA\leq AId$.
Then $||A||, ||A^-1||\leq \sqrt{K}$.
5. Area formula for the gradient of convex functions.
Let $\Omega$ be an open bounded set in $R^{n\times n}$, and let $u:\Omega\rightarrow R$ be a convex function of class $C^{1,1}_{loc}$. Then,
$|\nabla u(E)|=\int_E det（D^2u）dx , \forall E\subset \Omega Borel. $