Consider this group of six matrices:

Let G = <{I, A,B,C,D,K}, Matrix Multiplication>

I = $\displaystyle \left(\begin{array}{cc}1&0\\0&1\end{array}\right)$ A = $\displaystyle \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ B = $\displaystyle \left(\begin{array}{cc}0&1\\-1&-1\end{array}\right)$

C = $\displaystyle \left(\begin{array}{cc}-1&-1\\0&1\end{array}\right)$ D = $\displaystyle \left(\begin{array}{cc}-1&-1\\1&0\end{array}\right)$ K = $\displaystyle \left(\begin{array}{cc}1&0\\-1&-1\end{array}\right)$

Operation Table for this group:

_|I A B C D K

I |I A B C D K

A|A I C B K D

B|B K D A I C

C|C D K I A B

D|D C I K B A

K|K B A D C I

Define f: G -> <R*, $\displaystyle \bullet$> by y f(x) = det(x) for any Matrix x $\displaystyle /in$ G.

Question:

By which theorem in linear algebra, is f a Homomorphism from G to <R*, $\displaystyle \bullet$> ?