That is, if and then

Observe that the product
is defined if and only if

THE
NUMBER OF COLUMNS OF

For example, if
and
then

Observe the following:

- In this example, while
is defined,
the product
is not defined.
However, for square matrices and of the same order, both the product and are defined.

- The product
corresponds to operating on the
*rows of the matrix*(see 1.2.1), and - The product
also corresponds to operating on the
*columns of the matrix*(see 1.2.2).

- Note that if is a square matrix of order then Also, a scalar matrix of order commutes with any square matrix of order .
- In general, the matrix product is not commutative.
For example, consider the following two matrices
and
. Then check that the matrix
product

- Then That is, the matrix multiplication is associative.
- For any
- Then That is, multiplication distributes over addition.
- If is an matrix then
- For any square matrix
of order
and
we have
- the first row of is times the first row of
- for the row of is times the row of

Therefore,

Part 5. For all we have

as whenever Hence, the required result follows.

The reader is required to prove the other parts. height6pt width 6pt depth 0pt

- Let and be two matrices. If the matrix addition is defined, then prove that . Also, if the matrix product is defined then prove that .
- Let and Compute the matrix products and
- Let
be a positive integer. Compute
for the following matrices:
- Find examples for the following statements.
- Suppose that the matrix product is defined. Then the product need not be defined.
- Suppose that the matrix products and are defined. Then the matrices and can have different orders.
- Suppose that the matrices and are square matrices of order Then and may or may not be equal.

A K Lal 2007-09-12