= Solution
The determinant factors as
$$
\det A=-2(\mu-1)(\mu+1).
$$
Thus for $\mu\ne\pm1$, the <kernel of a linear map>[kernel] is $\{0\}$ and the inhomogeneous system has exactly one solution.
At $\mu=-1$, row reduction gives
$$
\ker A=\operatorname{span}\{(1,-2,3)^T\},
$$
and the augmented matrix has the same rank two as $A$, so there are infinitely many solutions. At $\mu=1$,
$$
\ker A=\operatorname{span}\{(-1,-2,3)^T\},
$$
but the augmented matrix has rank three while $A$ has rank two, so there is no solution. Therefore the number of solutions is
$$
\boxed{
\begin{cases}
1,&\mu\ne\pm1,\\
\infty,&\mu=-1,\\
0,&\mu=1.
\end{cases}}
$$
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