Scalar multiplication is the sort we learn about in grade school. It is one number multiplied by another. It is an alternative to repeated addition. Scalar multiplication is always commutative (a*b = b*a). Example 54* = 4*5 = 20 Matrix multiplication is an operation defined on rectangular arrays of numbers that meet certain requirements as to the array dimensions. A matrix of m rows and n columns may multiply one that is n rows and p columns. The values of m and p need not be anything in particular, and are often the same. The result of the multiplication is a matrix with m rows and p columns. In general, matrix multiplication is not commutative. (A•B ≠ B•A) Use of the term "number" in the above descriptions should be considered to include numeric constants, variables, and/or algebraic expressions. Example An example is found here.