Matrix/scalar multiplication is also analogous to vector/scalar multiplication, but matrix/matrix multiplication is somewhat more complicated. The greatest difficulty students have with matrix multiplication is that it can be a long process and therefore tedious. Since lots of arithmetic has to be done, there are ample opportunities to make simple mistakes. The process itself is straightforward. Take an 
matrix 
which represents the product of an 
matrix 
with a 
matrix 
. An entry 
is the dot product of the 
th row of 
(
) with the 
th column of 
(
). Using this definition iteratively for all the 
rows of 
and all the 
columns of 
will generate 
. This definition constrains the resulting dimensions of 
. Since the dot product is used, the number of columns in 
must equal the number of rows in 
so that the dotted vectors will have equal dimensions (in this case, both are equal to 
). Also, 
will have the same number of rows as 
(
) and the same number of columns as 
(
). This is easy to remember:
Additionally, matrix multiplication is not commutative: 
The reason is obvious when one looks back at the original definition. It is, however, both associative and distributive: 
matrix which represents the product of an matrix with a matrix . An entry is the dot product of the th row of () with the th column of (). Using this definition iteratively for all the rows of and all the columns of will generate . This definition constrains the resulting dimensions of . Since the dot product is used, the number of columns in must equal the number of rows in so that the dotted vectors will have equal dimensions (in this case, both are equal to ). Also, will have the same number of rows as () and the same number of columns as (). This is easy to remember:Excercise 3-2.
For each of the following matrix pairs, state whether or not the multiplication is possible (do not perform the multiplication). If the multiplication is valid, state the resulting dimensions:
and 
, Product Dimension: 
) 
and 
) 
and 
, Product Dimension: 
) 
and 
)