# 一、结论

对于矩阵$A=(a_{ij}(t))_{n\times n}$，其逆矩阵的导数为

$$\frac{dA^{-1}}{dt}=-A^{-1}\frac{dA}{dt}A^{-1}$$

# 二、证明过程

$$\begin{array}{l} \frac{dX^{-1}}{dx}=\lim_{\bigtriangleup x \to 0} \frac{\left ( X+\bigtriangleup X \right )^{-1}-X^{-1} }{\bigtriangleup x}\\ =\lim_{\bigtriangleup x \to 0} \frac{\left ( X+\bigtriangleup X \right )^{-1}XX^{-1}-\left ( X+\bigtriangleup X \right )^{-1}\left ( X+\bigtriangleup X \right )X^{-1}}{\bigtriangleup x}\\ =-\lim_{\bigtriangleup x \to 0} \left ( X+\bigtriangleup X \right )^{-1}\frac{\bigtriangleup X}{\bigtriangleup x} X^{-1}\\ =-X^{-1}\lim_{\bigtriangleup x \to 0} \frac{\bigtriangleup X}{\bigtriangleup x}X^{-1}\\ =-X^{-1}\frac{dX}{dx}X^{-1} \end{array}$$

$$\begin{array}{l} 0=I'=\left ( XX^{-1} \right ) '=X'X^{-1}+X\left ( X^{-1} \right )'\\ \Longrightarrow \left ( X^{-1} \right )'=-X^{-1}X'X^{-1} \end{array}$$

# 三、其它结论

不加证明的给出矩阵 A 的行列式的导数为

$$\frac{d\left | A \right | }{dt}=\left | A \right | tr\left (A^{-1} \frac{dA}{dt} \right )$$

其中 tr () 表示矩阵 A 的迹，即$tr(A)=a_{11}+a_{22}+ ... +a_{nn}\space$