Consider the sets and the composition laws that are defined as follows:

- The set of $2 \times 3$ matrices with real coefficients, and the composition law defined as the entry-by-entry matrix sum.
- The set of $2 \times 3$ matrices with real coefficients, and the composition law defined as the row-by-column matrix multiplication.
- The set of $2 \times 2$ matrices with real coefficients, and the composition law defined as the row-by-column matrix multiplication.
- The set of real numbers with the composition law $\odot$ defined by $$x \odot y=\frac{x}{y}$$
- The set of real, strictly positive numbers with the composition law $\circ$ defined by $$x \circ y=\frac{x}{y}$$
Check whether each of them is a magma, a semigroup, a monoid, a group, an Abelian group.