can a 3 by 4 matrix be multiplied by a 4 by 1 matrix

The general rule is that the number of columns of the first matrix must be equal to the number of rows of the second. To answer your question specifically, yes you
**can** multiply a 3x4 with a 4x1 but you **can't**
multiply a 4x1 with a 3x4. If you try it you'll see why, it's because you run out of entries to perform the operation on