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Recently Added Math Formulas
·
Curl of a Vector Field
·
Divergence of a Vector Field
·
Gradient of a Scalar Field
·
Properties of Transposes
·
The Transpose of a Matrix
Additional Formulas
·
Cartesian Coordinate
·
Cylindrical Coordinate
·
Spherical Coordinate
·
Transform from Cartesian to Cylindrical Coordinate
·
Transform from Cartesian to Spherical Coordinate
·
Transform from Cylindrical to Cartesian Coordinate
·
Transform from Spherical to Cartesian Coordinate
·
Divergence Theorem/Gauss' Theorem
·
Stokes' Theorem
·
Definition of a Matrix
Current Location
>
Math Formulas
>
Linear Algebra
> Properties of Transposes
Properties of Transposes
If
A = |a
ij
|
be a matrix of order m × n, then the matrix obtained by interchanging the rows and columns of
A
is known as the transpose of
A
. It is represented by
A
T
.
Hence if
A = |a
ij
|
of order
m × n, then
A
T
= |a
ij
|
of order
n × m.
Example:
If
, then
The following properties are valid for the transpose:
· The transpose of the transpose of a matrix is the matrix itself:
(A
T
)
T
= A
·
Transpose of a scalar multiple
:
The transpose of a matrix times a scalar (
k
) is equal to the constant times the transpose of the matrix:
(kA)
T
= kA
T
·
Transpose of a sum:
The transpose of the sum of two matrices is equivalent to the sum of their transposes:
(A + B)
T
= A
T
+ B
T
·
Transpose of a product:
The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order:
(AB)
T
= B
T
A
T
· The same is true for the product of multiple matrices:
(ABC)
T
= C
T
B
T
A
T
.
Example 1:
Find the transpose of the matrix
and verify that
(A
T
)
T
= A
.
Solution:
The transpose of matrix A is determined as shown below:
And the transpose of the transpose matrix is:
Hence
(A
T
)
T
= A.
Example 2:
If
and
, verify that
(A ± B)
T
= A
T
± B
T
.
Solution:
and the transpose of the sum is:
The transpose matrices for
A
and
B
are given as below:
And the sum of the transpose matrices is:
Hence
(A ± B)
T
= A
T
± B
T
.
Example 3:
If
and
, verify that
(AB)
T
= B
T
A
T
.
Solution:
The product of A and B is:
And the transpose of (AB) is:
If we take the transpose of A and B separately and multiply A with B, then we have:
Hence
(AB)
T
= B
T
A
T
.
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