Matrix multiplication

If necessary, refer to the matrix notation page for a refresher on the notation used to describe the sizes and entries of matrices.


Matrix-Scalar multiplication


The first kind of matrix multiplication is the multiplication of a matrix by a scalar, which will be referred to as matrix-scalar multiplication. A scalar is a number that makes things larger, smaller, or even negative (think of a negative scalar as "pointing backwards" or "flipping" something in the opposite direction). Given an m × n matrix, A, and a scalar, c, the matrix, cA, or Ac, is the m × n matrix whose i,jth entry is c times the i,jth entry of A. In other words,

for all i=1,...,m and j = 1,...,n. for example,


Properties of Matrix-Scalar multiplication


Matrix-Matrix multiplication


To multiply two matrices, A and B, the number of columns of A must equal the number of rows of B. For a product matrix, AB, to be defined in general, A must be m × n, and B must be n × p, for some positive integers m, n, and p. If A is m × n, adn B is n × p, the product matrix, AB, is the m × p matrix such that the i,jth entry of AB, denoted (AB)i,j, is the dot product of the ith row vector of A

[ai,1, ..., ai,n]

with the jth column vector of B

[b1,j, ..., bn,j]T

Therefore, (AB)i,j is given by the formula:

   = 
     = 
     = 

for all i = 1,2 ..., m and j = 1,2, ..., p (i runs through the rows of A, of which there are m, and j runs through the columns of B, of which there are p). For example, if A is the following 2 × 3 matrix and B is the following 3 × 2 matrix

then

   = 
     = 
     = 
     = 

corresponds to:

   = 
     = 
     = 

corresponds to:

   = 
     = 
     = 
     = 

corresponds to:

and

   = 
     = 
     = 
     = 

corresponds to:

Assembling these entries into the matrix product AB yields

AB    = 
     = 

However, if instead,

then AB is not defined since A has 3 columns and B has 2 rows, and 3 ≠ 2.


Properties of matrix multiplication