  In this paper, we introduce a ring isomorphism between the Clifford algebra
$C\ell_2$ and a ring of matrices, and represent the elements in $C\ell_2$ by
real matrices. By such a ring isomorphism, we introduce the concept of the
Moore-Penrose inverse in Clifford algebra $C\ell_2$. we solve the linear
equation $axb=d$, $ax=xb$ and $ax=\bar{x}b$. We also obtain necessary and
sufficient conditions for two numbers in $C\ell_2$ to be similar and
pseudosimilar.
