100 Amp Shed Workshop from 200 Amp House Panel 120' Direct Bury Cable
First, a few general thoughts:

You'll need four conductors, not three. Two hots, neutral, and a separate ground.
Also need to add a grounding electrode system at the outbuilding. (two ground rods, concrete-encased electrode, buried copper wire, copper water pipe, etc)
It seems short-sighted to direct-bury cable. You could put it in an enormously over-sized 2&quot; conduit for under US$1 per foot. 1-1/2 is adequate and, though it might be a tough pull, even 1-1/4 SCH 40 PVC meets the fill requirements for holding 3x1 ga (hot, hot, neutral) plus 1x4 ga (ground).
Don't forget to mount the riser conduits to the buildings (strut and conduit clamps, for example).
Some detached buildings provide prime roof real estate for PV generation. Does yours?  You might do well to drop an extra conduit in the trench so that PV wiring can be carried back from the outbuilding to the main panel in the future.
Will you ever want any low voltage connections between the buildings? Security, fire alarm, networking? If &quot;maybe&quot; then &quot;add another conduit.&quot;
Last rule of wiring: buy the wire last! Too often people buy the wire first, then partway through the project become aware of something that makes the already-bought wire less-ideal or even inappropriate/unusable.

I think your 2-2-4 cable is undersized. It's true that the 90 degree C column of NEC table 310.15(B)(16) pegs the ampacity of 2 ga aluminum at 100 A, but you can't use the 90 degree C column because your breakers' terminals will be rated for just 75 degree C.
The 2-2-4 cable sounds like a USE type cable? I'm not certain that's allowed to be used for a feeder. Also not sure that a feeder is allowed to down-size the neutral conductor the way services are.
For reference: I've just finished a 100 ft 100 A feeder similar to yours; I used 3x1 ga plus 1x4 ga aluminum THWN conductors in conduit. The 4 ga was for grounding, and although it could have been #8 copper or #6 aluminum, I used 4 ga because I needed a few for another circuit and because at that size and heavier one can re-mark a &quot;hot color&quot; conductor for neutral/ground use. I put a wrap of green tape at each end and that black wire suddenly became a ground wire.