  Given a representation $\varphi \colon B_n \to G_n$ of the braid group $B_n$,
$n \geq 2$ into a group $G_n$, we are considering the problem of whether it is
possible to extend this representation to a representation $\Phi \colon SM_n
\to A_n$, where $SM_n$ is the singular braid monoid and $A_n$ is an associative
algebra, in which the group of units contains $G_n$. We also investigate the
possibility of extending the representation $\Phi \colon SM_n \to A_n$ to a
representation $\widetilde{\Phi} \colon SB_n \to A_n$ of the singular braid
group $SB_n$. On the other hand, given two linear representations $\varphi_1,
\varphi_2 \colon H \to GL_m(\Bbbk)$ of a group $H$ into a general linear group
over a field $\Bbbk$, we define the defect of one of these representations with
respect to the other. Furthermore, we construct a linear representation of
$SB_n$ which is an extension of the Lawrence-Krammer-Bigelow representation
(LKBR) and compute the defect of this extension with respect to the exterior
product of two extensions of the Burau representation. Finally, we discuss how
to derive an invariant of classical links from the Lawrence-Krammer-Bigelow
representation.
