2x2-matrices/Determinant/Direct/Group homomorphism/Exercise

Let ${}K$ a field, and let

${}M={\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\mid a,b,c,d\in K,\,ad-bc\neq 0\right\}}\,$

the set of all invertible ${}2\times 2$-matrices.

a) Show that (without referring to the determinant), ${}M$ is, with matrix multiplication as operation, a group.

b) Show that (without referring to the determinant), the mapping

$M\longrightarrow K^{\times },{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\longmapsto ad-bc,$

is a group homomorphism.

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