Question about nodal analysis with an ideal op amp circuit
Your node A equation is incorrect because the voltage across R2 is \$V_A\$, not \$V_B\$. (It will turn out in this circuit that these voltages are the same, but for the purpose of doing nodal analysis, you can't assume this a priori)
Your node B equation is incorrect because you didn't include current coming from the op-amp output pin.
Your equation set is incorrect because you didn't include an equation for the output node.
The fact that it's difficult to write the equation for node B without knowing the internals of the op-amp, is why for hand calculation we don't usually use the nodal analysis. Instead we use equations derived from the knowledge that in negative feedback the two input terminals of the op-amp will be driven to the same voltage.