When i have a matrix

Question

4 X 4 and i have to multiply it by a
4 X 3
i know that the product size has to be a 4x 3

so do i do a row times column or how ?

It's easiest to learn using the notations for tensors. For any matrix X denote the entry in the i-th row and j-th column as:

X_{i,j}

Or, put differently, if you start at the top left corner and you take i steps downward and j steps to the right, you'll arrive at X_{i,j}

If X and Y are two matrices and Z = X Y, then the components of Z are given by:

Z_{i,j} = Sum over k of X_{i,k}Y_{k,j}

If X is the 4 by 4 matrix and Y is the 4 by 3 matrix, then the summation over k runs from 1 to 4, i can take values ranging from 1 to 4 and j can take values ranging from 1 to 3. Z is thus a 4 by 3 matrix.