前言
本文给出的这些公式是考研高等数学、线性代数和概率统计中常用且分数占比较高的公式，适合用于考前快速突击复习。
高等数学
01. 常用的等价无穷小公式
当 $x \rightarrow 0$ 时，有：
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \sin x \sim \tan x \sim \arcsin x \sim \arctan x \sim \mathrm{e}^{x} \textcolor{orangered}{-} 1 \sim \ln \left( 1+x \right) \sim x \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & a^{x} \textcolor{orangered}{-} 1 \sim x \ln a \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & 1 \textcolor{orangered}{-} \cos x \sim \frac{1}{2}x^{2} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( 1+x \right)^{a} \textcolor{orangered}{-} 1 \sim ax \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & x \textcolor{orangered}{-} \ln \left( 1+x \right) \sim \frac{1}{2}x^{2} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & x \textcolor{orangered}{-} \sin x \sim \arcsin x \textcolor{orangered}{-} x \sim \frac{1}{6}x^{3} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \tan x \textcolor{orangered}{-} x \sim x \textcolor{orangered}{-} \arctan x \sim \frac{1}{3}x^{3} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \tan x \textcolor{orangered}{-} \sin x \sim \arcsin x \textcolor{orangered}{-} \arctan x \sim \frac{1}{2}x^{3}
\end{aligned}
$$
02. 基本求导公式
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & C^{\prime} = 0, \ C \ \text{为常数} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( x^{\alpha} \right)^{\prime} = \alpha x^{\alpha-1} , \ \alpha \ \text{为常数} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \sin x \right)^{\prime} = \cos x \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \cos x \right)^{\prime} = \textcolor{orangered}{-} \sin x \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \tan x \right)^{\prime} = \sec^{2} x \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \cot x \right)^{\prime} = \textcolor{orangered}{-} \csc^{2} x \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \sec x \right)^{\prime} = \left( \sec x \right) \cdot \tan x \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \csc x \right)^{\prime} = \textcolor{orangered}{-} \left( \csc x \right) \cdot \cot x \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( a^{x} \right)^{\prime} = a^{x} \ln a , \ a \ \text{为常数} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \mathrm{e}^{x} \right)^{\prime} = \mathrm{e}^{x} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \log_{a} x \right)^{\prime} = \frac{1}{x \ln a} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \ln x \right)^{\prime} = \frac{1}{x} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \arcsin x \right)^{\prime} = \frac{1}{\sqrt{1-x^{2}}} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \arccos x \right)^{\prime} = \frac{\textcolor{orangered}{-}1}{\sqrt{1-x^{2}}} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \arctan x \right)^{\prime} = \frac{1}{1+x^{2}} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \text{arccot } x \right)^{\prime} = \frac{\textcolor{orangered}{-}1}{1+x^{2}}
\end{aligned}
$$
03. 莱布尼茨公式
若函数 $u \left( x \right)$, $v \left( x \right)$ 均 $n$ 阶可导，则：
$$
\left( uv \right)^{\left( n \right)} = \sum_{i=0}^{n} C_{n}^{i} u^{\left( i \right)} v^{\left( n-i \right)}
$$
其中 $u^{\left( 0 \right)} = u$, $v^{\left( 0 \right)} = v$, 且 $u = u(x)$, $v = v(x)$.
04. 常用的麦克劳林公式
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \mathrm{e}^{x} = 1 + x + \frac{x^{2}}{2!} + \cdots + \frac{x^{n}}{n!} + o \left( x^{n} \right) \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \sin x = x \textcolor{orangered}{-} \frac{x^{3}}{3!} + \cdots + \frac{\left( -1 \right)^{n} x^{2n+1}}{\left( 2n+1 \right)!} + o \left( x^{2n+1} \right) \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \cos x = 1 \textcolor{orangered}{-} \frac{x^{2}}{2!} + \cdots + \frac{\left( -1 \right)^{n} x^{2n}}{\left( 2n \right)!} + o \left( x^{2n} \right) \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \ln \left( 1 + x \right) = x \textcolor{orangered}{-} \frac{x^{2}}{2} + \frac{x^{3}}{3} + \cdots + \left( -1 \right)^{n-1} \frac{x^{n}}{n} + o \left( x^{n} \right) \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( 1 + x \right)^{m} = 1 + mx + \frac{m\left( m-1 \right)}{2!} x^{2} + \cdots + \frac{m\left( m-1 \right) \cdots \left( m-n+1 \right)}{n!} x^{n} + o \left( x^{n} \right)
\end{aligned}
$$
05. 函数渐近线
$\textcolor{lightgreen}{\blacktriangleright}$ 水平渐近线：
若 $\lim_{x \rightarrow +\infty} f \left( x \right) = b$ 或 $\lim_{x \rightarrow -\infty} f \left( x \right) = b$，则 $y = b$ 为函数 $y = f \left( x \right)$ 的水平渐近线.
$\textcolor{lightgreen}{\blacktriangleright}$ 垂直渐近线：
若 $\lim_{x \rightarrow x_{0}^{-}} f \left( x \right) = \infty$ 或 $\lim_{x \rightarrow x_{0}^{+}} f \left( x \right) = \infty$，则 $x = x_{0}$ 为函数 $y = f \left( x \right)$ 的垂直渐近线.
$\textcolor{lightgreen}{\blacktriangleright}$ 斜渐近线：
若 $k = \lim_{x \rightarrow +\infty} \frac{f \left( x \right)}{x}$, $b = \lim_{x \rightarrow +\infty} \left[f \left( x \right) \textcolor{orangered}{-} kx \right]$, 或者 $k = \lim_{x \rightarrow -\infty} \frac{f \left( x \right)}{x}$, $b = \lim_{x \rightarrow -\infty} \left[f \left( x \right) \textcolor{orangered}{-} kx \right]$，则直线 $y = kx + b$ 是曲线 $y = f \left( x \right)$ 的斜渐近线.
06. 曲率和曲率半径公式
如果在直角坐标下，曲线 $y = f \left( x \right)$ 二阶可导，则曲率为：
$$
k = \frac{\left|y^{\prime\prime}\right|}{\left( 1 + y^{\prime 2} \right)^{\frac{3}{2}}}
$$
如果曲线由参数方程 $\begin{cases} x = \varphi \left( t \right) \ y = \psi \left( t \right) \end{cases}$ 确定，且 $\varphi \left( t \right)$, $\psi \left( t \right)$ 二阶可导，则曲率为：
$$
k = \frac{\left| \varphi^{\prime} \left( t \right) \psi^{\prime\prime} \left( t \right) \textcolor{orangered}{-} \varphi^{\prime\prime}\left( t \right)\psi^{\prime} \left( t \right) \right|}{\left[\varphi^{\prime 2} \left( t \right) + \psi^{\prime 2} \left( t \right) \right]^{\frac{3}{2}}}
$$
曲率半径为：
$$
R = \frac{1}{k}, \text{ 其中 } k \neq 0
$$
07. 基本积分公式
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \int x^{k} \mathrm{~d}x = \frac{x^{k+1}}{k+1} + C, \ k \neq -1 \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{x} \mathrm{~d}x = \ln \left| x \right| + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int a^{x} \mathrm{~d}x = \frac{a^{x}}{\ln a} + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \mathrm{e}^{x} \mathrm{~d}x = \mathrm{e}^{x} + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \cos x \mathrm{~d}x = \sin x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \sin x \mathrm{~d}x = \textcolor{orangered}{-} \cos x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{\sin x} \mathrm{~d}x = \int \csc x \mathrm{~d}x = \ln \left| \csc x – \cot x \right| + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{\cos x} \mathrm{~d}x = \int \sec x \mathrm{~d}x = \ln \left| \sec x + \tan x \right| + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{\sin^{2} x} \mathrm{~d}x = \int \csc^{2} x \mathrm{~d}x = \textcolor{orangered}{-} \cot x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{\cos^{2} x} \mathrm{~d}x = \int \sec^{2} x \mathrm{~d}x = \tan x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \tan x \mathrm{~d}x = \textcolor{orangered}{-} \ln \left| \cos x \right| + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \cot x \mathrm{~d}x = \ln \left|\sin x\right| + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \left( \sec x \right) \cdot \tan x \mathrm{~d}x = \sec x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \left( \csc x \right) \cdot \cot x \mathrm{~d}x = -\csc x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{a^{2} + x^{2}} \mathrm{~d}x = \frac{1}{a} \arctan \frac{x}{a} + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{1 + x^{2}} \mathrm{~d}x = \arctan x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{\sqrt{a^{2} – x^{2}}} \mathrm{~d}x = \arcsin \frac{x}{a} + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{\sqrt{1 – x^{2}}} \mathrm{~d}x = \arcsin x + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{a^{2} – x^{2}} \mathrm{~d}x = \frac{1}{2a} \ln \left|\frac{a + x}{a – x}\right| + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{1 – x^{2}} \mathrm{~d}x = \frac{1}{2} \ln \left|\frac{1 + x}{1 – x}\right| + C \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \int \frac{1}{\sqrt{x^{2} \pm a^{2}}} \mathrm{~d}x = \ln \left|x + \sqrt{x^{2} \pm a^{2}}\right| + C
\end{aligned}
$$
线性代数
01. 数字型行列式的常用公式
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ &
\begin{vmatrix}
a_{11} & 0 & \cdots & 0 \\
0 & a_{22} & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & a_{nn} \end{vmatrix} =
\begin{vmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
0 & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & a_{nn} \end{vmatrix} =
\begin{vmatrix} a_{11} & 0 & \cdots & 0 \\
a_{21} & a_{22} & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix} =
a_{11} a_{22} \cdots a_{nn} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ &
\begin{vmatrix}
0 & & & \lambda_{1} \\
& & \lambda_{2} & \\
& \cdots & & \\
\lambda_{n} & & & 0
\end{vmatrix}
\begin{vmatrix}
0 & & & \lambda_{1} \\
& & \lambda_{2} & * \\
& \cdots & \vdots & \vdots \\
\lambda_{n} & \cdots & * & *
\end{vmatrix}
\begin{vmatrix}
* & \cdots & * & \lambda_{1} \\
* & \cdots & \lambda_{2} & \\
\vdots & \cdots & & \\
\lambda_{n} & & & 0
\end{vmatrix}
= (-1)^{\frac{n(n-1)}{2}} \lambda_{1} \lambda_{2} \cdots \lambda_{n}
\end{aligned}
$$
设 $\boldsymbol{A}$ 是 $m$ 阶方阵，$\boldsymbol{B}$ 是 $n$ 阶方阵，则：
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B} \end{vmatrix} = \begin{vmatrix} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{vmatrix} = \begin{vmatrix} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{vmatrix} = \left| \boldsymbol{A} \right|\left| \boldsymbol{B} \right| \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{vmatrix} = \begin{vmatrix} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{C} \end{vmatrix} = \begin{vmatrix} \boldsymbol{C} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{vmatrix} = \left( \textcolor{orangered}{-} 1 \right)^{mn}\left| \boldsymbol{A} \right|\left| \boldsymbol{B} \right|
\end{aligned}
$$
范德蒙行列式：
$$
\textcolor{lightgreen}{\blacktriangleright} \ \boldsymbol{D}_{n} = \begin{vmatrix} 1 & 1 & 1 & \cdots & 1 \\ x_{1} & x_{2} & x_{3} & \cdots & x_{n} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & \cdots & x_{n}^{2} \\ \vdots & \vdots & \vdots & & \vdots \\ x_{1}^{n \textcolor{orangered}{-} 1} & x_{2}^{n \textcolor{orangered}{-} 1} & x_{3}^{n \textcolor{orangered}{-} 1} & \cdots & x_{n}^{n \textcolor{orangered}{-} 1} \end{vmatrix} = \prod_{1 \leqslant j < i \leqslant n} \left( x_{i} \textcolor{orangered}{-} x_{j} \right)
$$
02. 抽象型行列式的常用公式
设 $\boldsymbol{A}$, $\boldsymbol{B}$ 为 $n$ 阶方阵，则：
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\boldsymbol{A}^{\top}
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\lambda \boldsymbol{A}
\end{vmatrix} = \lambda^{n} \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\boldsymbol{A} \boldsymbol{B}
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{B} \boldsymbol{A}
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} \begin{vmatrix}
\boldsymbol{B}
\end{vmatrix} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\boldsymbol{A}^{k}
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix}^{k} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\boldsymbol{A}^{\textcolor{orangered}{-}1}
\end{vmatrix} = \frac{1}{\begin{vmatrix}
\boldsymbol{A}
\end{vmatrix}}, \ \text{若 } \boldsymbol{A} \text{ 可逆} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\boldsymbol{A}^{*}
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix}^{n \textcolor{orangered}{-} 1}, \ n \geqslant 2 \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} = \lambda_{1}\lambda_{2} \cdots \lambda_{n}, \text{其中 } \lambda_{1}, \lambda_{2}, \cdots, \lambda_{n} \text{ 是 } \boldsymbol{A} \text{ 的 } n \text{ 个特征值} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \text{若 } \boldsymbol{A} \text{ 与 } \boldsymbol{B} \text{ 相似}，\text{ 则 } \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{B}
\end{vmatrix}
\end{aligned}
$$
03. 克拉默法则
$\textcolor{lightgreen}{\blacktriangleright}$ 如果线性方程组 $\begin{cases} a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} = b_{2} \\ \vdots \\ a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{nn}x_{n} = b_{n} \end{cases}$ 的系数行列式 $\boldsymbol{D} = \begin{vmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{n1} & \cdots & a_{nn} \end{vmatrix} \neq 0$，则该方程组有唯一解：
$$
x_{1} = \frac{\boldsymbol{D}_{1}}{\boldsymbol{D}}, \ x_{2} = \frac{\boldsymbol{D}_{2}}{\boldsymbol{D}}, \ \cdots, \ x_{n} = \frac{\boldsymbol{D}_{n}}{\boldsymbol{D}}
$$
其中 $\boldsymbol{D}_{j}$ $\left( j = 1,2,\cdots,n \right)$ 是把系数行列式 $\boldsymbol{D}$ 中第 $j$ 列用常数项代替后所得的 $n$ 阶行列式.
04. 矩阵的转置
矩阵的转置满足下列运算规律：
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \boldsymbol{A}^{\top} \right)^{\top} = \boldsymbol{A} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \boldsymbol{A} + \boldsymbol{B} \right)^{\top} = A^{\top} + B^{\top} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \lambda \boldsymbol{A} \right)^{\top} = \lambda \boldsymbol{A}^{\top} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \boldsymbol{A}\boldsymbol{B} \right)^{\top} = \boldsymbol{B}^{\top} \boldsymbol{A}^{\top}
\end{aligned}
$$
05. 伴随矩阵
伴随矩阵的性质：
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \boldsymbol{A} \boldsymbol{A}^{*} = \boldsymbol{A}^{*} \boldsymbol{A} = \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} \boldsymbol{E} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( k \boldsymbol{A} \right)^{*} = k^{n \textcolor{orangered}{-} 1} \boldsymbol{A}^{*}, \ n \geqslant 2 \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \boldsymbol{A} \boldsymbol{B} \right)^{*} = \boldsymbol{B}^{*} \boldsymbol{A}^{*} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \begin{vmatrix}
\boldsymbol{A}^{*}
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix}^{n \textcolor{orangered}{-} 1}, \ n \geqslant 2 \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \boldsymbol{A}^{*} \right)^{\textcolor{orangered}{-} 1} = \left( \boldsymbol{A}^{\textcolor{orangered}{-} 1} \right)^{*} = \frac{1}{\begin{vmatrix}
\boldsymbol{A}
\end{vmatrix}} \boldsymbol{A} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \boldsymbol{A}^{*} \right)^{\top} = \left( \boldsymbol{A}^{\top} \right)^{*} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left( \boldsymbol{A}^{*} \right)^{*} = \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix}^{n \textcolor{orangered}{-} 2} \boldsymbol{A}, \ n \geqslant 3 \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \text{一般情况下 } \left( \boldsymbol{A} + \boldsymbol{B} \right)^{*} \neq \boldsymbol{A}^{*} + \boldsymbol{B}^{*}
\end{aligned}
$$
06. 可逆及不可逆的充分必要条件
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ n \text{ 阶矩阵 } \boldsymbol{A} \text{ 可逆 } & \Leftrightarrow \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} \neq 0 \\ \\
& \Leftrightarrow \boldsymbol{A} \boldsymbol{B} = \boldsymbol{E} \\ \\
& \Leftrightarrow \boldsymbol{B} \boldsymbol{A} = \boldsymbol{E} \\ \\
& \Leftrightarrow \mathrm{r} \left( \boldsymbol{A} \right) = n \\ \\
& \Leftrightarrow \boldsymbol{A}^{*} \text{ 可逆} \\ \\
& \Leftrightarrow \boldsymbol{A} \text{ 可以表示为若干个初等矩阵的乘积} \\ \\
& \Leftrightarrow \boldsymbol{A} \text{ 与 } \boldsymbol{E} \text{ 等价} \\ \\
& \Leftrightarrow \boldsymbol{A} \boldsymbol{x} = 0 \text{ 只有零解} \\ \\
& \Leftrightarrow \forall \boldsymbol{b}, \ \boldsymbol{A} \boldsymbol{x} = \boldsymbol{b} \text{ 有唯一解} \\ \\
& \Leftrightarrow \boldsymbol{A} \text{ 的列（行）向量组线性无关} \\ \\
& \Leftrightarrow \boldsymbol{A} \text{ 的特征值都不为 } 0
\end{aligned}
$$
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ n \text{ 阶矩阵 } \boldsymbol{A} \text{ 不可逆 } & \Leftrightarrow \begin{vmatrix}
\boldsymbol{A}
\end{vmatrix} = 0 \\ \\
& \Leftrightarrow \mathrm{r} \left( \boldsymbol{A} \right) < n \\ \\
& \Leftrightarrow \boldsymbol{A} \boldsymbol{x} = 0 \text{ 有非零解} \\ \\
& \Leftrightarrow \boldsymbol{A} \text{ 的列（行）向量组线性相关} \\ \\
& \Leftrightarrow 0 \text{ 是 } \boldsymbol{A} \text{ 的特征值}
\end{aligned}
$$
07. 可逆矩阵的性质
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ & \text{若 } \boldsymbol{A} \text{ 可逆}, \text{ 则 } \boldsymbol{A}^{-1} \text{ 亦可逆}, \text{ 且 } \left( \boldsymbol{A}^{-1} \right)^{-1} = \boldsymbol{A} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \text{若 } \boldsymbol{A} \text{ 可逆}, \text{ 则 } k \boldsymbol{A} \text{ 亦可逆}, \text{ 且 } \left( kA \right)^{-1} = \frac{1}{k}A^{-1}, \text{ 其中 } k \neq 0 \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \text{若 } \boldsymbol{A}, \boldsymbol{B} \text{ 可逆}, \text{ 则 } \boldsymbol{A} \boldsymbol{B} \text{ 亦可逆}, \text{ 且 } \left( \boldsymbol{A} \boldsymbol{B} \right)^{-1} = \boldsymbol{B}^{-1} \boldsymbol{A}^{-1} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \text{若 } \boldsymbol{A} \text{ 可逆}, \text{ 则 } \boldsymbol{A}^{\top} \text{ 亦可逆}, \text{ 且 } \left( \boldsymbol{A}^{\top} \right)^{-1} = \left( \boldsymbol{A}^{-1} \right)^{\top} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \left|A^{-1}\right| = \frac{1}{\left|A\right|} \\ \\
\textcolor{lightgreen}{\blacktriangleright} \ & \text{一般情况下 } \left( A + B \right)^{-1} \neq A^{-1} + B^{-1}
\end{aligned}
$$
概率统计
01. 事件的运算律
$\textcolor{lightgreen}{\blacktriangleright}$ 交换律：$A \cup B = B \cup A$, $A \cap B = B \cap A$
$\textcolor{lightgreen}{\blacktriangleright}$ 结合律：$A \cup \left( B \cup C \right) = \left( A \cup B \right) \cup C$, $\ A \cap \left( B \cap C \right) = \left( A \cap B \right) \cap C$
$\textcolor{lightgreen}{\blacktriangleright}$ 分配律：$A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left( A \cup C \right)$, $A \cap \left( B \cup C \right) = \left( A \cap B \right) \cup \left( A \cap C \right)$
$\textcolor{lightgreen}{\blacktriangleright}$ 摩根律（对偶律）：$\overline{A \cup B} = \overline{A} \cap \overline{B}$, $\overline{A \cap B} = \overline{A} \cup \overline{B}$
02. 概率的计算公式
$\textcolor{lightgreen}{\blacktriangleright}$ 加法公式：
$$
\begin{aligned}
& P \left( A \cup B \right) \\
= \ & P \left( A \right) + P \left( B \right) \textcolor{orangered}{-} P \left( AB \right) \\ \\
& P \left( A \cup B \cup C \right) \\
= \ & P \left( A \right) + P \left( B \right) + P \left( C \right) + P \left( ABC \right) \textcolor{orangered}{-} P \left( AB \right) \textcolor{orangered}{-} P \left( BC \right) \textcolor{orangered}{-} P \left( AC \right)
\end{aligned}
$$
$\textcolor{lightgreen}{\blacktriangleright}$ 减法公式：
$$
P \left( B \textcolor{orangered}{-} A \right) = P \left( B \right) \textcolor{orangered}{-} P \left( AB \right)
$$
$\textcolor{lightgreen}{\blacktriangleright}$ 乘法公式：
若 $P \left( A \right) > 0$, 则 $P \left( AB \right) = P \left( B|A \right)P \left( A \right)$;
若 $P \left( B \right) > 0$, 则 $P \left( AB \right) = P \left( A|B \right) P \left( B \right)$;
若 $P \left( AB \right) > 0$, 则 $P \left( ABC \right) = P \left( C|AB \right)P \left( B|A \right) P \left( A \right) = P \left( C|AB \right)P \left( A|B \right)P\left( B \right)$.
$\textcolor{lightgreen}{\blacktriangleright}$ 全概率公式：
$$
P \left( A \right) = \sum_{i=1}^{n} P \left( A|B_{i} \right) P\left( B_{i} \right)
$$
其中 $B_{i}B_{j} = \varnothing$, $\bigcup_{i=1}^{n} B_{i} = \Omega$, 其中 $i \neq j$, 且事件 $B_{i}$ 的个数可以是可列个.
$\textcolor{lightgreen}{\blacktriangleright}$ 贝叶斯公式：
$$
P\left( B_{j}|A \right) = \frac{P\left( A|B_{j} \right)P\left( B_{j} \right)}{\sum_{i=1}^{n} P\left( A|B_{i} \right)P\left( B_{i} \right)}
$$
其中 $B_{i}B_{j} = \varnothing$, $\bigcup_{i=1}^{n} B_{i} = \Omega$, 其中 $i \neq j$, 且事件 $B_{i}$ 的个数可以是可列个.
03. 事件的独立性
$\textcolor{lightgreen}{\blacktriangleright}$ $A$ 与 $B$ 独立 $\Leftrightarrow$ $P \left( AB \right) = P\left( A \right)P\left( B \right)$.
$\textcolor{lightgreen}{\blacktriangleright}$ $A$, $B$, $C$ 两两独立 $\Leftrightarrow$ $\begin{cases} P \left( AB \right) = P \left( A \right) P \left( B \right), \\ P \left( AC \right) = P \left( A \right)P \left( C \right), \\ P \left( BC \right) = P \left( B \right) P \left( C \right). \end{cases}$
$\textcolor{lightgreen}{\blacktriangleright}$ $A$, $B$, $C$ 相互独立 $\Leftrightarrow$ $\begin{cases} P \left( AB \right) = P \left( A \right) P \left( B \right), \\ P \left( BC \right) = P \left( B \right)P \left( C \right), \\ P \left( AC \right) = P \left( A \right) P \left( C \right), \\ P \left( ABC \right) = P \left( A \right) P \left( B \right) P \left( C \right). \end{cases}$
04. 事件独立的性质及相关结论
$\textcolor{lightgreen}{\blacktriangleright}$ 若事件 $A$, $B$ 相互独立，则 $A$ 与 $\overline{B}$, $\overline{A}$ 与 $B$, $\overline{A}$ 与 $\overline{B}$ 也相互独立.
$$
\begin{aligned}
\textcolor{lightgreen}{\blacktriangleright} \ \text{事件} A, B \text{ 独立} & \Leftrightarrow P\left( AB \right) = P \left( A \right) P \left( B \right) \\ \\
& \Leftrightarrow P \left( B \right) = P \left( B|A \right) \\ \\
& \Leftrightarrow P \left( B \right) = P \left( B|\overline{A} \right) \\ \\
& \Leftrightarrow P \left( B|A \right) = P \left( B|\overline{A} \right)
\end{aligned}
$$
其中，$0 < P \left( A \right) < 1$.
$\textcolor{lightgreen}{\blacktriangleright}$ 若 $A_{1}$, $A_{2}$, $\cdots$, $A_{m}$, $B_{1}$, $B_{2}$, $\cdots$, $B_{n}$ 相互独立，则 $f \left( A_{1},A_{2},\cdots,A_{m} \right)$ 与 $g \left( B_{1},B_{2},\cdots,B_{n} \right)$ 也相互独立，其中 $f \left( \cdot \right)$, $g \left( \cdot \right)$ 分别表示对相应事件作任意事件运算.
$\textcolor{lightgreen}{\blacktriangleright}$ 若 $P \left( A \right) = 0$ 或 $P \left( A \right) = 1$, 则 $A$ 与任何事件 $B$ 都相互独立.
05. 独立、互斥、互逆之间的关系
$\textcolor{lightgreen}{\blacktriangleright}$ $A$ 与 $B$ 互逆 $\Rightarrow$ $A$ 与 $B$ 互斥，但反之不一定成立；
$\textcolor{lightgreen}{\blacktriangleright}$ $A$ 与 $B$ 互斥（或互逆）且均为非零概率事件 $\Rightarrow$ $A$ 与 $B$ 不独立；
$\textcolor{lightgreen}{\blacktriangleright}$ $A$ 与 $B$ 相互独立且均为非零概率事件 $\Rightarrow$ $A$ 与 $B$ 不互斥；
$\textcolor{lightgreen}{\blacktriangleright}$ 一般情况下，独立和互斥无关，独立推不出互斥、互斥也推不出独立.
06. 分布函数的性质
$\textcolor{lightgreen}{\blacktriangleright}$ 非负性：$0 \leqslant F \left( x \right) \leqslant 1$；
$\textcolor{lightgreen}{\blacktriangleright}$ 规范性：$f \left( \textcolor{orangered}{-} \infty \right) = 0$, $f \left( + \infty \right) = 1$；
$\textcolor{lightgreen}{\blacktriangleright}$ 单调不减性： 对于任意 $x_{1} < x_{2}$，有 $f \left( x_{1} \right) \leqslant f \left( x_{2} \right)$；
$\textcolor{lightgreen}{\blacktriangleright}$ 右连续性： $F \left( x_{0} + 0 \right) = F \left( x_{0} \right)$.
07. 密度函数的性质
$\textcolor{lightgreen}{\blacktriangleright}$ 非负性： $f \left( x \right) \geqslant 0$, 其中 $-\infty < x < +\infty$;
$\textcolor{lightgreen}{\blacktriangleright}$ 规范性：$\int_{- \infty}^{+ \infty} f \left( x \right)\mathrm{~d}x = 1$；
$\textcolor{lightgreen}{\blacktriangleright}$ 对于任意实数 $a$ 和 $b$, 有 $P \left\{a < X \leqslant b\right\} = \int_{a}^{b} f \left( x \right) \mathrm{~d} x$, 其中 $a < b$;
$\textcolor{lightgreen}{\blacktriangleright}$ 对于连续型随机变量 $X$, 有 $P\left\{ X = x \right \} = 0$, 对 $\forall x \in R$ 成立；
$\textcolor{lightgreen}{\blacktriangleright}$ 连续型随机变量的分布函数 $F \left( x \right)$ 是连续函数；
$\textcolor{lightgreen}{\blacktriangleright}$ 在 $f \left( x \right)$ 的连续点处，有 $F^{\prime}\left( x \right) = f \left( x \right)$.
高等数学
涵盖高等数学基础概念、解题技巧等内容，图文并茂，计算过程清晰严谨。
线性代数
以独特的视角解析线性代数，让繁复的知识变得直观明了。
特别专题
通过专题的形式对数学知识结构做必要的补充，使所学知识更加连贯坚实。
让考场上没有难做的数学题！