线性代数公式汇总

行列式

$\begin{aligned} & \large |A^T|=|A|; \quad\quad\quad\quad |kA|=k^n|A|;\\ & \large |AB|=|A||B|; \quad\quad\quad\quad |A^*|=|A|^{n-1};\\ & \large |A^{-1}|=|A|^{-1}=\frac{1}{|A|}; \quad\quad\quad\quad A\sim B \longrightarrow |A|=|B|.\\ \end{aligned}$

矩阵

转置矩阵

$\begin{aligned} & \large (A^T)^T=A; \quad\quad\quad\quad (A+B)^T=A^T+B^T; \\ & \large (kA)^T=kA^T; \quad\quad\quad\quad (AB)^T=B^TA^T.\\ \end{aligned}$

逆矩阵

$\begin{aligned} & \large (A^{-1})^{-1}=A; \quad\quad\quad\quad (kA)^{-1}=\frac{1}{k}A^{-1};\\ & \large (AB)^{-1}=B^{-1}A^{-1}; \quad\quad\quad\quad (A^n)^{-1}=(A^{-1})^n;\\ & \large (A^{-1})^T=(A^T)^{-1}; \quad\quad\quad\quad A^{-1}=\frac{1}{|A|}A^*.\\ \end{aligned}$

伴随矩阵

$\begin{aligned} & \large AA^*=A^*A=|A|E; \quad\quad\quad\quad A^*=|A|A^{-1};\\ & \large |A^*|=|A|^{n-1}; \quad\quad\quad\quad (A^*){-1}=(A^{-1})^*=\frac{1}{|A|}A;\\ & \large (A^*)^T=(A^T)^*; \quad\quad\quad\quad (kA)^*=k^{n-1}A^*;\\ & \large (A^*)^*=|A|^{n-2}A;\\ \\ & \large r(A^*)= \begin{cases} n, \quad 若r(A)=n, \\ 1, \quad 若r(A)=n-1,\\ 0, \quad 若r(A)<n-1. \end{cases} \end{aligned}$

矩阵的秩

$\begin{aligned} & \large r(A)=r(A^T)=r(AA^T)=r(A^TA);\\ & \large r(A+B)\leqslant r(A)+r(B); \quad\quad\quad\quad r(AB)\leqslant min\{r(A),r(B)\};\\ & \large 若P、Q可逆，r(A)=r(PA)=r(AQ)=r(PAQ);\quad若不可逆，r(AQ)<r(A);\\ & \large 若A为m\times n矩阵，B为n\times s矩阵，且AB=O，则r(A)+r(B)\leqslant n.\\ \end{aligned}$

相关性无关性

$\begin{aligned} & \large A=(\alpha_1,\alpha_2,……,\alpha_m)\\ & \large 向量组A线性相关\Longleftrightarrow \begin{cases} \end{cases} \end{aligned}$