Inverse z-transform. Where is mistake?
$$X_1(z) = \frac{2.6z^{-1} - 1.8z^{-2}}{1 - 2.3z^{-1} + 1.2z^{-2}} =\frac{A}{1 - 1.5z^{-1}} + \frac{B}{1 - 0.8z^{-1}} \tag{1}$$


  Mistake is that $A \neq 2$ and $B \neq -0.5$. 


Well then, what are they ? There aren't any $A,B$ where the above equation is true. Why? Take the limit $z \rightarrow \infty$. The term $\frac{A}{1 - 1.5z^{-1}} + \frac{B}{1 - 0.8z^{-1}}$ factors to maximally $z^{-1}$ in the numerator whereas $\frac{2.6z^{-1} - 1.8z^{-2}}{1 - 2.3z^{-1} + 1.2z^{-2}} $ contains a $z^{-2}$ term!!!!. So a contradiction !! The correct decomposition is as follows:
$$X_1(z) = \frac{2.6z^{-1} - 1.8z^{-2}}{1 - 2.3z^{-1} + 1.2z^{-2}} =\frac{A}{1 - 1.5z^{-1}} + \frac{B+Cz^{-1}}{1 - 0.8z^{-1}}$$

Taking a common denominator in $\frac{A}{1 - 1.5z^{-1}} + \frac{B}{1 - 0.8z^{-1}}$, equation $(1)$ is now equivalent to
$$2.6z^{-1} - 1.8z^{-2} =A+B + (C -1.5B - 0.8)z^{-1} -1.5Cz^{-1}$$
By identification, we get the following system of equations
$$\begin{bmatrix}
    1.0000 &amp;   1.0000   &amp;      0\\
         0 &amp;  -1.5000  &amp;  1.0000\\
         0  &amp;       0  &amp; -1.5000
\end{bmatrix}
\begin{bmatrix}
A\\B\\C
\end{bmatrix}
=
\begin{bmatrix}
         0\\
    3.4000\\
   -1.8000
\end{bmatrix}
$$
which will give you
\begin{align}
A &amp;= 1.4667\\
B &amp;=  -1.4667 \\
C &amp;= 1.2
\end{align}
Can you continue ?