Suppose $GL(n,\mathbb{R})$ is a multiplicative group of invertible matrix ( $n$x $n$). Let $\mathbb{R}$ be an additive group of real numbers. Given $g:GL(n,\mathbb{R})\rightarrow\mathbb{R}$ with $g(A)=tr(A)$.
Is $g$ homomorphism? If yes then prove it!
For g to be a homomorphism, we need that for every $A,B \in GL_n(\mathbb{R}), tr(A)tr(B) = tr(AB)$