You may be uneasy with mathematical operations being defined in this casual fashion. Here are some more accurate definitions:

Matrix times Column Matrix:LetAbe a M x C matrix, andxbe a C x 1 column matrix. Then the productAxis the M x 1 column matrix whosei-thelement is the dot product of row i ofAwithx.

Row Matrix times Matrix:Letxbe a 1 x R row matrix andAbe a R x M matrix, Then the productxAis the 1 x M row matrix whosei-thelement is the dot product ofxwith column i ofA.