Since my knowledge of matrix algebra is limited… Suppose I have several transformations of the same object - a translate, a rotate and a scale, for example - instead transforming the objects three times, can I just multiply the transforms and then transform once?

It’s quite logical really: if you translate first, and then scale (TS) then the translation is scaled as well. If you scale first and then translate (ST) the translation happens after scaling and is not scaled.

You are probably aware of this, but another important thing to keep in mind is that if you are going to rotate your object in place, it must first be translated back to the origin, rotated, then translated back to it’s original location. This is because rotations always take place around the origin and if you leave the object in it’s place away from the origin it will orbit the origin.