The product
of two Matrices
and
is defined by

(1) 
where is summed over for all possible values of and . Therefore, in order for multiplication to be defined, the
dimensions of the Matrices must satisfy

(2) 
where denotes a Matrix with rows and columns. Writing out the product explicitly,

(3) 
where
Matrix multiplication is Associative, as can be seen by taking

(4) 
Now, since , , and are Scalars, use the Associativity
of Scalar Multiplication to write

(5) 
Since this is true for all and , it must be true that

(6) 
That is, matrix multiplication is Associative. However, matrix multiplication is not, in general,
Commutative (although it is Commutative if
and
are Diagonal and
of the same dimension).
The product of two Block Matrices is given by multiplying each block





(7) 
See also Matrix, Matrix Addition, Matrix Inverse, Strassen Formulas
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 178179, 1985.
© 19969 Eric W. Weisstein
19990526