Basic arithmetic with matrices We have matrices. 
$$A=\begin{bmatrix}
-2 & 0 \\
-5 & 6 \\
\end{bmatrix}
B^{-1}=\begin{bmatrix}
-7 & 8 \\
2 & -8 \\
\end{bmatrix}
C=\begin{bmatrix}
-15 & -2 \\
-8 & -14 \\
\end{bmatrix}
$$
We need to solve matrix $X$ from equation:
$$A^{-1}XB-C=0$$
$$X=AB^{-1}+C$$
$$X=\begin{bmatrix}
-2 & 0 \\
-5 & 6 \\
\end{bmatrix}
\begin{bmatrix}
-7 & 8 \\
2 & -8 \\
\end{bmatrix}
+
\begin{bmatrix}
-15 & -2 \\
-8 & -14 \\
\end{bmatrix}
$$
$$
X=\begin{bmatrix}
-1 & -18 \\
39 & -102 \\
\end{bmatrix}
$$
This doesn't seem to be correct solutions to this equations ?

As point out by you that the order matters:
Pre-multiply and post-multiply are different.
Starting from
$${A^{ - 1}}XB - C = 0$$
Add $C$ on both sides
$${A^{ - 1}}XB = C$$
Now post-multiply by $B^{-1}$ on both sides
$${A^{ - 1}}X{B}{B^{ - 1}} = C{B^{ - 1}}$$
$${A^{ - 1}}X{I} = C{B^{ - 1}}$$
$${A^{ - 1}}X = C{B^{ - 1}}$$
Now pre-multiply by $A$ on both sides to get the final form
$${A}{A^{ - 1}}X = AC{B^{ - 1}}$$
$$X = AC{B^{ - 1}}$$
