Matrix operations are performed on the whole matrix. A matrix has usually at least 2 x 2 dimensions, a vector has 1 x p or n x 1 dimensions, and a scalar has 1 x 1 dimensions and is therefore a single number. Scalar multiplication is easy.
Matrix multiplication is unusual.
The new elements result from the sums of pairwise multiplications of the original elements, according to an orderly procedure. These are the dimensions across which the summing is performed. 1x1 Matrix multiplication is associative and distributive, but not commutative. This unit vector multiplication performs summing operations on matrices. Thus, any matrix that is postmultiplied by a weighting vector results in a new vector of numbers that are the weighted sums performed across the matrix's columns.