第一章 行列式

设A、B为n阶矩阵

$$\left|A^T\right|=\left|A\right|$$

$$\left|A^m\right|={\left|A\right|}^m$$

$$\left|kA\right|=k^n\left|A\right|$$

$$\left|AB\right|=\left|A\right|\left|B\right|$$

$$若A可逆，则\left|A^{-1}\right|=\frac{1}{\left|A\right|}$$

$$\left|A^{\ast }\right|={\left|A\right|}^{n-1}$$

$$A{A}^{\ast }=A^{\ast }A=\left|A\right|E$$

$$A^{\ast }=\left|A\right|A^{-1}(若A可逆)$$

$$A=\left|A\right|(A^{\ast }{)}^{-1}$$

$$\left|\begin{array}{ccc}{A}_1& & \\ & A_2& \\ & & A_3\end{array}\right|=A_1{A}_2{A}_3,\left|\begin{array}{ccc}& & A_1\\ & A_2& \\ A_3& & \end{array}\right|=-A_1{A}_2{A}_3$$

设A为n阶矩阵，B为m阶矩阵，根据拉普拉斯展开定理有

$$\left|\begin{array}{cc}A& 0\\ 0& B\end{array}\right|=\left|\begin{array}{cc}A& C\\ 0& B\end{array}\right|=\left|\begin{array}{cc}A& 0\\ C& B\end{array}\right|=\left|A\right|\left|B\right|$$

$$\left|\begin{array}{cc}0& A\\ B& 0\end{array}\right|=\left|\begin{array}{cc}C& A\\ B& 0\end{array}\right|=\left|\begin{array}{cc}0& A\\ B& C\end{array}\right|=(-1{)}^{mn}\left|A\right|\left|B\right|$$

化“叉”型行列式

$$\left|\begin{array}{ccccc}a& 0& ...& 0& b\\ ...& & A& & ...\\ c& 0& ...& 0& d\end{array}\right|=(ad-bc)\left|A\right|,其中A是方阵,且除了主对角线和副对角线以外其余所有的元素均为0$$

化“ab”型行列式

$$\left|\begin{array}{ccccc}a& b& b& ...& b\\ b& a& b& ...& b\\ b& b& a& ...& b\\ ...& ...& ...& ...& b\\ b& b& b& ...& a\end{array}\right|=[a+(n-1)b](a-b{)}^{n-1}$$

特征值求行列式

$$若题干可求得矩阵A的所有特征值{\lambda }_1,{\lambda }_2...,{\lambda }_n,那么立即有\left|A\right|={\lambda }_1{\lambda }_2...{\lambda }_n$$

第二章 矩阵

矩阵转置的性质

$$(A^T{)}^T=A$$

$$(kA{)}^T=k{A}^T$$

$$(A\pm B{)}^T=A^T\pm B^T$$

$$(AB{)}^T=B^T{A}^T$$

$$(A^{-1}{)}^T=(A^T{)}^{-1}$$

$$(A^T{)}^m=(A^m{)}^T$$

矩阵伴随的性质

$$A^{\ast }=\left|A\right|A^{-1}(若A可逆)$$

$$A^{-1}=\frac{1}{\left|A\right|}{A}^{\ast }$$

$$(A^T{)}^{\ast }=(A^{\ast }{)}^T$$

$$(kA{)}^{\ast }=k^{n-1}{A}^{\ast }$$

$$(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$$

$${\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}^{\ast }=\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$$