Let
be a general matrix and let
denote a
general matrix. Denote the matrix product by
. Then *matrix multiplication* is carried out by computing
the *inner product* of every row of
with every column of
. Let the th row of
be denoted by
,
, and the th column of
by
,
. Then the matrix product
is
defined as

**Examples:**

An matrix
can be multiplied on the *right* by an
matrix, where is any positive integer. An
matrix
can be multiplied on the *left* by a
matrix, where is any positive integer. Thus, the number of columns in
the matrix on the left must equal the number of rows in the matrix on the
right.

Matrix multiplication is *non-commutative*, in general. That is,
normally
even when both products are defined (such as when the
matrices are square.)

The *transpose of a matrix product* is the product of the
transposes in *reverse order*:

The *identity matrix* is denoted by
and is defined as

As a special case, a matrix times a vector produces a new vector which consists of the inner product of every row of with

As a further special case, a row vector on the left may be multiplied by a
column vector on the right to form a *single inner product*:

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