How is the positive input of this op-amp calculated
I now want to calculate the positive voltage input. The gain of the amplifier is -(R3/R1) = -9. \$ V_{out} \$  is calculated by
\$ V_{out} = A(v^+ -   v^-) \$

In this formula, the \$A\$ is the open-loop gain of the op-amp, not the closed loop gain of the op-amp circuit with feedback.
\$A\$ should be 100,000 or so, not 9.
However, we wouldn't normally use this formula to solve the circuit.
We'd simply assume \$A\$ is &quot;very large&quot;. From this (and the presence of negative feedback) we can get the rule that
$$v^+ = v^-$$
Then, the rule that (very nearly) no current flows into the op-amp's input pins, tells us that the current through R3 is the same as the current through R1.
Then we have
$$\frac{v_{in} -v^-}{1000} = \frac{v^- - v_{out}}{9000}$$
and we can solve to get the final answer.