Page 137 - 35Linear Algebra

7.3 Properties of Matrices 137

This rule obeys linearity.
Notice that in order for the multiplication to make sense, the columns
and rows must match. For an r × k matrix M and an s × m matrix N, then
to make the product MN we must have k = s. Likewise, for the product
NM, it is required that m = r. A common shorthand for keeping track of
the sizes of the matrices involved in a given product is the following diagram.

r × k times k × m is r × m

Reading homework: problem 2

Example 84 Multiplying a (3×1) matrix and a (1×2) matrix yields a (3×2) matrix.

     
1 1 · 2 1 · 3 2 3

3 2 3 = 3 · 2 3 · 3 = 6 9 .
     
2 2 · 2 2 · 3 4 6
Another way to view matrix multiplication is in terms of dot products:

The entries of MN are made from the dot products of the rows of
M with the columns of N.

Example 85 Let

   T 
1 3 u
2 3 1
M =  3 5  =:  v T  and N = 0 1 0 =: a b c
2 6 w T
where

1 3 2 2 3 1
u = , v = , w = , a = , b = , c = .
3 5 6 0 1 0
Then
   
u · a u · b u · c 2 6 1
MN =  v · a v · b v · c  =  6 14 3  .
w · a w · b w · c 4 12 2

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