Back muscles are not adequate in themselves to supply the needed torso rigidity. Evolution has left us with musculature designed for quadripedal animals, and the muscles which could support a hanging, horizontal spine donít stabilize a vertical one subjected to all the impacts and forces that upright posture dictates.
However, we have abdominal musculature which can aid in torso support. These are the muscles which contract the body to enable a running animal to bring its legs forward. The quads straighten the leg. The hamstrings bend it at the knee. The abdominal and groin muscles pull the leg over the top of the pedal stroke. But they do something else, too. They provide stiffness to the torso to support and reflect the force of the legs, whether pushing away against the ground in running, or pushing against the pedals in riding a bicycle. This is where we get stability on the bike.
If your only strength work is on your abdominals, this alone will vastly improve your riding. Strong abdominals are also the key to preserving a healthy back, and are the foundation of strength for a strong rider. Riding with undeveloped abdominals is something like riding a bike with a cracked frame. All the energy gets dissipated in flexion, and doesn't get you down the road.
Squats are also a good exercise to strengthen the torso, if done right. The weight and torso should rise together as the legs straighten, with the abdominals tight, helping the back muscles to hold the torso rigid. If you can't lift with this style, you are lifting too heavy a weight.