How does one multiply a row vector by a matrix?

What you wrote would be the result of multiplying A*C^T, where C^T is the COLUMN vector C-transpose. For C*A you want c_1*a_11 + c_2*a_21 + c_3*a_31 + c_4*a_41. Notice that C*A is a 1X4 matrix, while A*C^T is 4X1.