Product of Matrices
Product of Matrices,
Let
,
and
,
for matrix
A
of order m x n,and matrix
B
of order p x q, product of matrices exist if and only if
. Then the product of matrices is
where the order of matrix
D
is x by y and
,
.
The elements of
D are
Or
For example,
From the definition of matrix multiplication, it is possible to have
without
or
and
without
or
.
Properties of Matrix Multiplication
If both matrix B
premultiplied by matrix
A,
and matrix
B
postmultipled by matrix
A,
are defined,
. In general, matrix multiplication is not commutative.
. Multiplication of matrices is associative.
For matrix
A
of order m x n, matrix
B
of order p x q and matrix
C
of order s x t, product of matrices exist if and only if
and 
. Then element of
is
And the element of
is
Rearrange the order of terms
And equals to the element of
or
.
Matrix multiplication is distributive with respect to addition.
If
, then
and
.
If A
and D
are of same order and
B
and C
are square matrices, then
A,
B,
C
and D
are all square matrices.
Since
equals
and the element is
And equals to the element of
.
Identity Matrix
The identity under matrix multiplication is the identity matrix, that is
Let
,
and
then
and
In order to have the same order as
A,
for element x, order of k should equal to i and for
element y, order of l should equal to j, and therefore
the identity matrix must be square.
According to the equality of matrices,
only when
,
and
only when
.
The form of identity matrix is
.
If
, that is
, then
A
must be a square matrix.
And therefore it is also called unit matrix.
Inverse Matrix
If A
has an multiplication inverse
and
then
A
must be square.
If a matrix has an inverse, the inverse is unique.
Assume both B
and C
are inverses of A,
then
and
.
Premultiply first equation by
C,
imply
.
From second equation, imply
and the inverse is unique.
Division of matrices has not been defined, the multiplying of the
reciprocal of a matrix can be replaced by multiplying the inverse of the matrix.
Therefore solving
B
in
can be obtained by premultiplying
.
Let
, If both matrices have an inverse, then
. That is
The inverse of a matrix's inverse equals to the matrix itself,
. Let
then
Transpose of a matrix
Transpose of a matrix,
or
The transpose of a matrix, written as
, is obtained by interchanging the rows and
columns of the original matrix
A.
Therefore if
amd
, the elemnt,
of matrix
equals to the element,
of matrix,
A.
And if the order of
A
is m x n then the order of
is n x
m.
For example,
Properties of Matrix Transpose
, if and only if
.
Matrix Products Transpose
if the matrix multiplication of
exists, then
.
Since
is defined,
is also defined while
may not be.
Let
, where
A
is of order m x n and
B
is of order n x r, and
C
is of order m x r then
and element of
C is
Let D
is the transpose of
C,
then
That is element in the row j and colume
i of matrix
D
equals to product of row i of matrix
A
and column j of matrix
B,
that is
Let
, since
is of order n x
m and
is of order r x
n, then
E
is of order r x m.
Let X
be the transpose of
A, imply
,
Let Y
be the transpose of
B,
imply
and imply
,
That is element in the row j and colume
i of matrix
E
equals to
,
Subsitute corresponding elements in row of matrix
B
and in column of matrix
A
imply
,
And equals to the element of
.
Therefore
Transpose of Inverse Matrix
If
exists, then
.
Since
and
, imply
,
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