Let $G=M_{3\times 3}(\mathbb{Z})$, let $*$ denote componentwise multiplication of matrices, then-
(a) $(G,*)$ is a group
(b) $(G,*)$ is a monoid
(c) $G$ has exactly $2^9$ elements having inverse
(d) $G$ has exactly $4^9$ elements having inverse
Explanation: Available soon.
Explanation 1
Explanation 2
Explanation 3
Explanation 4
Other explanation.