Electronic – Why does LTSpice say that the “Matrix is singular” for this ideal-transformer circuit

Many circuit simulators attempt to find a DC operating point prior to executing a transient analysis. Basically, they replace inductors with short circuits and capacitors with open circuits and then solve the remaining circuit.

Your circuit has no stable DC point (open-circuit current source) and this process fails. Usually, there's a way to tell the simulator to skip the initialization step and specify starting conditions (e.g., Vcap = 0V) explicitly.

You only need to consider the phase around the feedback loop. Since you don't have feedback from the output of the second to the input of the first you can treat the two independently.

If you wanted to feedback from the output back to the input you would have problems.

You have rather low feedback resistors - is there a reason for that? They will consume a large amount of power and of your output current capability just to drive the feedback.

It will help reduce Johnson noise but is not really necessary in the second stage as noise there will be dwarfed by that in the first stage.