Textbook Problem

Let $G$ be the set of all elements of $M_2\left(ℝ\right)$ that have one row that consists of zeros and one row of the form $\left[aa\right],$ with $a\ne 0$.

Show that $G$ is closed under multiplication in $M_2\left(ℝ\right)$.

Show that for each $x$ in $G,$ there is an element $y$ in $G$ such that $xy=yx=x$.

Show that $G$ does not have an identity element with respect to multiplication.