OME 1972 Problem 1

Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$ . An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of $M$ that have inverse they are characterized by the condition $ad - bc \ne 0$ . c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.