Question

If A $=\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$ $B=\left[\begin{array}{cc}c& d\\ -d& c\end{array}\right]$ show that A product B commute

i. e. AB = BA also verify that

$|AB|=|A|.|B|$

Answer 1

Here we can easily show that

AB = BA $=\left[\begin{array}{cc}ac-bd& ad-bd\\ -bc-ad& -bd-ac\end{array}\right]$

$\therefore DetAB=|AB|(ac-bd{)}^2+(ac-bc{)}^2$

$=a^2{c}^2+b^2{d}^2+a^2{d}^2+b^2{c}^2{c}^2$

$=(a^2+b^2(c^2+d^2)=|A||B|$ = Det A . Det B