u • v = [u1v1, u2v2, ...]
The dot product gives us a scalar, not another vector. The products are added together, not put into vector components.
Matrix multiplication involves taking dot products of rows from A with columns from B to get the respective row,column entry in AB
1/A, where A is a matrix
There is no such thing as dividing a scalar by a matrix. Division by a matrix does not exist, and what you're probably thinking of/looking for is the inverse of the matrix, A-1.