Matrix

1. Matrix computing attributes

   $\begin{array}{l}AB\ne BA\\ A(BC)=(AB)C\\ A(B+C)=AB+AC\\ (A+B)C=AC+BC\\ {(AB)}^T=B^T{A}^T\\ |(AB)|=|A||B|\\ A^n=AA\mathrm{...}A\\ A^0=I\\ {(\alpha A)}^n={\alpha }^n{A}^n\\ |A^n|={|A|}^n\\ I_n=\left[\begin{array}{cccc}1& \mathrm{...}& 0& 0\\ 0& 1\mathrm{...}& \mathrm{...}& 0\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ 0& 0& \mathrm{...}& 1\end{array}\right]\end{array}$

2. Matrix addition

   $\begin{array}{l}A\pm B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\pm \left[\begin{array}{cc}x& y\\ z& u\end{array}\right]=\left[\begin{array}{}a& \pm xb& \pm y\\ c& \pm zd& \pm u\end{array}\right]\end{array}$

3. Matrix multiplication

   $\begin{array}{l}xA=x\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{}x& ax& b\\ x& cx& d\end{array}\right]=\left[\begin{array}{}a& xb& x\\ c& xd& x\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]x=Ax\\ \left[\begin{array}{cc}a& b\end{array}\right]\left[\begin{array}{cc}x& y\\ z& u\end{array}\right]=\left[\begin{array}{}ax+bzay+bu\end{array}\right]\\ \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}x& y\\ z& u\end{array}\right]=\left[\begin{array}{}ax+bzay+bu\\ cx+dzcy+du\end{array}\right]\\ AB=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]\left[\begin{array}{ccc}x& y& z\end{array}\right]=\left[\begin{array}{ccc}ax& bx& cx\\ ay& by& cy\\ az& bz& cz\end{array}\right]\\ BA=\left[\begin{array}{ccc}x& y& z\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}ax+by+cz\end{array}\right]\\ CB=\left[\begin{array}{ccc}a& b& c\\ p& q& r\\ u& v& w\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}ax+by+cz\\ px+qy+rz\\ ux+vy+wz\end{array}\right]\\ BD=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{ccc}j& k& l\\ m& n& o\\ p& q& r\end{array}\right]=\left[\begin{array}{ccc}aj+bm+cp& ak+bn+cq& al+bo+cr\\ dj+em+fp& dk+en+fq& dl+eo+fr\\ gj+hm+ip& gk+hn+iq& gl+ho+ir\end{array}\right]\\ DB=\left[\begin{array}{ccc}j& k& l\\ m& n& o\\ p& q& r\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]=\left[\begin{array}{ccc}ja+kd+lg& jb+ke+lh& jc+kf+li\\ ma+nd+og& mb+ne+oh& mc+nf+oi\\ pa+qd+rg& pb+qe+rh& pc+qf+ri\end{array}\right]\\ AB\ne BA\end{array}$

4. Determinant

   $\begin{array}{l}\left[\begin{array}{}A\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=(ad-bc)\\ \left[\begin{array}{}A\end{array}\right]=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]=a\left[\begin{array}{cc}e& f\\ h& i\end{array}\right]-b\left[\begin{array}{cc}d& f\\ g& i\end{array}\right]+c\left[\begin{array}{cc}d& e\\ g& h\end{array}\right]=a(ei-fh)-b(di-fg)+c(dh-eg)=aei+bfg+cdh-ceg-bdi-afh\end{array}$

5. Transposed matrix

   $\begin{array}{l}A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\\ A^T=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\\ {\left[\begin{array}{cc}a& b\\ c& d\\ u& v\end{array}\right]}^T=\left[\begin{array}{ccc}a& c& u\\ b& d& v\end{array}\right]\end{array}$

6. Inverse matrix

   $\begin{array}{l}{A}^{-1}=\frac{1}{|A|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ A^{-1}={\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]}^{-1}=\frac{1}{|A|}{\left[\begin{array}{ccc}A& B& C\\ D& E& F\\ G& H& I\end{array}\right]}^T=\frac{1}{|A|}\left[\begin{array}{ccc}A& D& G\\ B& E& H\\ C& F& I\end{array}\right]\\ |A|=a(ei-fh)-b(id-fg)+c(dh-eg)\\ A=(ei-fh)\\ B=-(di-fg)\\ C=(dh-eg)\\ D=-(bi-ch)\\ E=(ai-cg)\\ F=-(ah-bg)\\ G=(bf-ce)\\ H=-(af-cd)\\ I=(ae-bd)\end{array}$