3.I.1A

Given two real non-zero $2 \times 2$ matrices $A$ and $B$, with $A B=0$, show that $A$ maps $\mathbb{R}^{2}$ onto a line. Is it always true that $B A=0 ?$ Show that there is always a non-zero matrix $C$ with $C A=0=A C$. Justify your answers.

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