## Matrix Multiplication

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)